','

D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)

{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}

{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">

$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$

{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Needs review

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
Note: This page needs review to confirm formulas and sources.
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Needs review

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
Note: This page needs review to confirm formulas and sources.
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}\]
  
{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}


  
Formula (extracted LaTeX)
  
\[{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">\]
  
{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">


  
Formula (extracted text)
  
  
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$


  
Formula (extracted text)
  
  
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$


  
Formula (extracted text)
  
  
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$

      
Variables and units
      

        
No variables provided in audit spec.
      
    
  

  

    Sources (authoritative):
    

      
No sources provided in audit spec.
    
  

  

    Changelog

    

      Version: 0.1.0-draft

      Last code update: 2026-01-19
    
    

  0.1.0-draft · 2026-01-19
  

    
Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.
  

  

  

    Verification pending · Last code update: 2026-01-19

lt;/p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Needs review

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
Note: This page needs review to confirm formulas and sources.
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Needs review

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
Note: This page needs review to confirm formulas and sources.
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}\]
  
{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}


  
Formula (extracted LaTeX)
  
\[{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">\]
  
{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">


  
Formula (extracted text)
  
  
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$


  
Formula (extracted text)
  
  
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$


  
Formula (extracted text)
  
  
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$

      
Variables and units
      

        
No variables provided in audit spec.
      
    
  

  

    Sources (authoritative):
    

      
No sources provided in audit spec.
    
  

  

    Changelog

    

      Version: 0.1.0-draft

      Last code update: 2026-01-19
    
    

  0.1.0-draft · 2026-01-19
  

    
Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.
  

  

  

    Verification pending · Last code update: 2026-01-19

lt;/p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Complete

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Needs review

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
Note: This page needs review to confirm formulas and sources.
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    






    





    

      
    

    

      
      
      
      

        
        

          

            

              
Distance Calculator (Euclidean Coordinate Distance)
              
Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below.
              

              

                
Endpoints $P_1$ and $P_2$
                
                

                    
Point
                    
X-Coordinate
                    
Y-Coordinate
                    
Z-Coordinate (Optional)
                    
                    
P₁
                    
                    
                    
                    
                    
P₂
                    
                    
                    
                
                
Leave Z coordinates blank for a 2D calculation.
                
                

                  Calculate Distance
                
                
                


                

                  
Results
                  
                  

                    

                      
Distance ($D$)
                      

                    
                    

                      
Midpoint ($M$)
                      

                    
                  

                  
Step-by-Step Solution
                  

                    
                
              
              
              
The Euclidean Distance Formula
              
The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$).

              
Distance Formula (2D)
              
Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
              

              
Distance Formula (3D)
              
For three dimensions, the formula simply adds the difference in the $z$ coordinates:
              

                $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$
              
              
              
Related Formula: Midpoint
              
The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value:
              

                $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$
              

              
Frequently Asked Questions (FAQ)
              

                

                  
What is the Euclidean Distance Formula (2D)?
                  

                    
The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
                  
                
                

                  
How does the distance formula relate to the Pythagorean Theorem?
                  

                    
The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added.
                  
                
                

                  
What is the difference between Euclidean and Geodesic distance?
                  

                    
Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy.
                  
                
                

                  
Can the distance calculator handle negative coordinates?
                  

                    
Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance.
                  
                
              

            
 : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}


  Audit: Needs review

  

    
Formula (LaTeX) + variables + units
    

      This section shows the formulas used by the calculator engine, plus variable definitions and units.
      
Note: This page needs review to confirm formulas and sources.
    
    

      

  
Formula (extracted LaTeX)
  
\[','\]
  
','


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}


  
Formula (extracted LaTeX)
  
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
  
D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}


  
Formula (extracted LaTeX)
  
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)\]
  
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)


  
Formula (extracted LaTeX)
  
\[{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}\]
  
{is3D ? `<br>$\\Delta z = z_2 - z_1 = ${formatNumber(p2.z)} - ${formatNumber(p1.z)} = ${formatNumber(dz)}$` : ''} </p> <p>2. <strong>Square and Sum the Differences:</strong></p> <p class="formula-box text-center"> $(\\Delta x)^2 = ${formatNumber(dx_sq)}$<br> $(\\Delta y)^2 = ${formatNumber(dy_sq)}


  
Formula (extracted LaTeX)
  
\[{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">\]
  
{is3D ? ` + ${formatNumber(dz_sq)}` : ''} = ${formatNumber(sum_sq)}$ </p> <p>3. <strong>Calculate Distance ($D$):</strong> Take the square root of the sum.</p> <p class="formula-box text-center"> $D = \\sqrt{\\sum} = \\sqrt{${formatNumber(sum_sq)}} \\approx ${formatNumber(distance)}$ </p> <h4 class="font-semibold text-lg mt-4">Midpoint Calculation:</h4> <p>Midpoint X: $\\frac{x_1 + x_2}{2} = \\frac{${formatNumber(p1.x)} + ${formatNumber(p2.x)}}{2} = ${formatNumber(Mx)}$</p> <p>Midpoint Y: $\\frac{y_1 + y_2}{2} = \\frac{${formatNumber(p1.y)} + ${formatNumber(p2.y)}}{2} = ${formatNumber(My)}$</p> ${is3D ? `<p>Midpoint Z: $\\frac{z_1 + z_2}{2} = \\frac{${formatNumber(p1.z)} + ${formatNumber(p2.z)}}{2} = ${formatNumber(Mz)}$</p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">


  
Formula (extracted text)
  
  
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$


  
Formula (extracted text)
  
  
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$


  
Formula (extracted text)
  
  
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$

      
Variables and units
      

        
No variables provided in audit spec.
      
    
  

  

    Sources (authoritative):
    

      
No sources provided in audit spec.
    
  

  

    Changelog

    

      Version: 0.1.0-draft

      Last code update: 2026-01-19
    
    

  0.1.0-draft · 2026-01-19
  

    
Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.
  

  

  

    Verification pending · Last code update: 2026-01-19

lt;/p>` : ''} `; // 4. Update DOM const midpointCoords = `(${formatNumber(Mx)}, ${formatNumber(My)}${is3D ? `, ${formatNumber(Mz)}` : ''})`; resultDistance.textContent = formatNumber(distance); resultMidpoint.textContent = midpointCoords; stepByStep.innerHTML = steps; resultsContainer.classList.remove('hidden'); // Re-render MathJax if (window.MathJax) { window.MathJax.typesetPromise([stepByStep]); } } catch (e) { showError(e.message); } }); // Initial calculation (for example values) calculateBtn.click(); }); </script> </div> </main> <aside class="w-full lg:w-1/3"> <div class="sticky top-28 space-y-6"> <div class="bg-white p-6 rounded-lg shadow-md"> <h3 class="text-xl font-semibold mb-4">Core Formulas</h3> <div class="text-center font-bold text-gray-700 space-y-4"> <div class="p-3 bg-blue-50 rounded-lg"> <p class="text-sm text-gray-600">Distance (2D)</p> <p class="text-xl text-blue-600">