Equilateral Triangle Calculator
Compute side length, height, area, perimeter, inradius and circumradius of an equilateral triangle from any one known value.
Interactive Equilateral Triangle Solver
All linear dimensions will use the same unit.
Results
Note: All lengths are shown in the selected unit. Area is shown in squared units.
Equilateral Triangle Diagram
In an equilateral triangle, all sides are equal (s), all angles are 60°, and the height (h) is also an altitude, median, and angle bisector. This symmetry leads to the compact formulas used in the calculator.
Equilateral triangle formulas
An equilateral triangle is fully determined by any one of its key measurements. Once you know one of side, height, area, perimeter, inradius, or circumradius, you can compute all the others.
From side length \(s\)
Perimeter
\[ P = 3s \]
Height (altitude)
\[ h = \frac{\sqrt{3}}{2}\,s \]
Area
\[ A = \frac{\sqrt{3}}{4}\,s^2 \]
Inradius (inscribed circle)
\[ r = \frac{\sqrt{3}}{6}\,s = \frac{h}{3} \]
Circumradius (circumscribed circle)
\[ R = \frac{\sqrt{3}}{3}\,s = \frac{2h}{3} = 2r \]
From height \(h\)
Side length
\[ s = \frac{2}{\sqrt{3}}\,h \]
Then use the side-based formulas above to get \(P, A, r, R\).
From area \(A\)
Side length
\[ s = \sqrt{\frac{4A}{\sqrt{3}}} \]
Again, once you have \(s\), everything else follows.
From perimeter \(P\)
Side length
\[ s = \frac{P}{3} \]
From inradius \(r\)
Side length
\[ s = \frac{6}{\sqrt{3}}\,r = 2\sqrt{3}\,r \]
From circumradius \(R\)
Side length
\[ s = \frac{3}{\sqrt{3}}\,R = \sqrt{3}\,R \]
How the equilateral triangle formulas are derived
Take an equilateral triangle with side length \(s\). Drop a height from the top vertex to the base. This splits the triangle into two congruent right triangles with:
- Hypotenuse \(s\)
- Base \(s/2\)
- Height \(h\)
By the Pythagorean theorem:
\[ h^2 + \left(\frac{s}{2}\right)^2 = s^2 \quad\Rightarrow\quad h^2 = s^2 - \frac{s^2}{4} = \frac{3}{4}s^2 \quad\Rightarrow\quad h = \frac{\sqrt{3}}{2}s \]
Then the area is:
\[ A = \frac{1}{2}\cdot \text{base} \cdot \text{height} = \frac{1}{2}\cdot s \cdot \frac{\sqrt{3}}{2}s = \frac{\sqrt{3}}{4}s^2 \]
The inradius and circumradius follow from standard triangle relations, but for an equilateral triangle they simplify nicely to:
\[ r = \frac{A}{\tfrac{1}{2}P} = \frac{A}{\tfrac{3}{2}s} = \frac{\sqrt{3}}{6}s, \qquad R = 2r = \frac{\sqrt{3}}{3}s \]
Worked example
Suppose you have an equilateral triangle with side length \(s = 10\ \text{cm}\). Then:
- Perimeter: \(P = 3s = 30\ \text{cm}\)
- Height: \(h = \frac{\sqrt{3}}{2}\cdot 10 \approx 8.6603\ \text{cm}\)
- Area: \(A = \frac{\sqrt{3}}{4}\cdot 10^2 \approx 43.3013\ \text{cm}^2\)
- Inradius: \(r = \frac{\sqrt{3}}{6}\cdot 10 \approx 2.8868\ \text{cm}\)
- Circumradius: \(R = 2r \approx 5.7735\ \text{cm}\)
The calculator performs these steps instantly and keeps units consistent.
Common questions
Do units matter in the formulas?
Yes, but only in the usual way:
- If you enter a length in centimeters, all other lengths come out in centimeters.
- Area is always in squared units (cm², m², in², ft², etc.).
Can a triangle be equilateral and isosceles?
Yes. An equilateral triangle is a special case of an isosceles triangle where all three sides (not just two) are equal.
When should I use this instead of a general triangle calculator?
Use this tool whenever you know the triangle is equilateral. The dedicated formulas are simpler, more stable numerically, and let you solve the triangle from a single measurement instead of needing multiple sides or angles.