Distance Calculator (Euclidean Coordinate Distance)
Calculate the straight-line distance between two points ($P_1, P_2$) in 2D or 3D space using the Euclidean distance formula. Includes step-by-step solution and midpoint calculation.
Endpoints $P_1$ and $P_2$
Leave z-coordinates blank for a 2D measurement.
How to Use This Calculator
Enter the Cartesian coordinates of both endpoints. Provide z-coordinates whenever you are measuring a 3D vector; leave them blank for planar distances. Click Calculate or edit any field to refresh the result.
Methodology
This calculator applies fixed Euclidean geometry: differences between coordinates become the legs of a right triangle, the squared sums give the squared hypotenuse, and a square root finishes the distance. The midpoint is just the average of corresponding coordinates.
- Step 1: Subtract each coordinate to find $\Delta x$, $\Delta y$, and optional $\Delta z$.
- Step 2: Square the differences and add them together.
- Step 3: You get the distance by taking the square root of the sum.
- Bonus: Midpoint coordinates are just $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})$ in 3D.
Frequently Asked Questions
What is the Euclidean Distance Formula (2D)?
The 2D formula finds the straight-line length between $P_1(x_1, y_1)$ 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?
It is the Pythagorean Theorem applied to coordinate differences: the legs are $\Delta x$ and $\Delta y$ (plus $\Delta z$), and the hypotenuse is the distance.
What is the difference between Euclidean and Geodesic distance?
Euclidean distance is straight-line distance on a flat plane. Geodesic distance smoothly follows curved surfaces, like Earth, often using great-circle calculations.
Can the calculator handle negative inputs?
Yes. Squaring the differences removes the sign, so negative coordinates behave identically to positives.