Why does the area formula use √3 / 4?

The height of an equilateral triangle is (√3 / 2)a, and half of base times height gives ½ × a × (√3 / 2)a = (√3 / 4)a². The constant is therefore fixed.

How do I find the inradius and circumradius?

Both scale with the side: r = a√3 / 6 ≈ 0.2887a and R = a / √3 ≈ 0.5774a. The circumradius is exactly twice the inradius.

Is a triangle with three 60° angles always equilateral?

Yes. Equal angles force equal opposite sides, so specifying all three angles or all three sides describes the same unique shape.

📐 Equilateral Triangle Calculator

Calculate area, height and perimeter of an equilateral triangle from side length.

Input Method Side Length

What is an Equilateral Triangle?

A triangle where all three sides are equal and all interior angles are 60°. Knowing any one value allows you to calculate all others.

When side length (a) is known

h = (√3 / 2) × a

L = 3a

S = (√3 / 4) × a² = (1/2) × a × h

When height (h) is known

a = 2h / √3

L = 2√3 × h

S = (1/√3) × h²

When area (S) is known

h = √(√3 × S)

a = √(4S / √3)

L = 3a

Properties of an Equilateral Triangle

In an equilateral triangle, all four centers (circumcenter, incenter, centroid, orthocenter) coincide. Circumradius R = a/√3, inradius r = a/(2√3).

Worked examples

A side of 6 gives an area of (√3 / 4) × 36 = 9√3 ≈ 15.5885, a height of 3√3 ≈ 5.1962 and a perimeter of 18.

A side of 10 gives an area of 25√3 ≈ 43.3013 with a height of 5√3 ≈ 8.6603 and a perimeter of 30.

What the result assumes

All three sides are equal and every interior angle is exactly 60°, so one measurement fixes the shape completely. The height is also the median and the angle bisector, and the area grows with the square of the side: doubling 6 to 12 quadruples the area to about 62.3538. Keep the input in one unit to read area in square units of the same unit.

Frequently asked questions

Why does the area formula use √3 / 4?: The height of an equilateral triangle is (√3 / 2)a, and half of base times height gives ½ × a × (√3 / 2)a = (√3 / 4)a². The constant is therefore fixed.
How do I find the inradius and circumradius?: Both scale with the side: r = a√3 / 6 ≈ 0.2887a and R = a / √3 ≈ 0.5774a. The circumradius is exactly twice the inradius.
Is a triangle with three 60° angles always equilateral?: Yes. Equal angles force equal opposite sides, so specifying all three angles or all three sides describes the same unique shape.