Equilateral Triangle Calculator

Last updated: 2026-05-18

Definition

An equilateral triangle is a triangle whose three sides are equal in length and whose three interior angles are each exactly 60°. Because every side and angle is determined by a single measurement, the side length alone fixes the height, area, perimeter, circumradius, and inradius.

Height

Drop a perpendicular from any vertex to the opposite side. By symmetry it bisects that side, splitting the equilateral triangle into two congruent 30-60-90 triangles. Using the Pythagorean theorem:

$h^2 = a^2 - \left(\frac{a}{2}\right)^2 = \frac{3a^2}{4} \implies h = \frac{a\sqrt{3}}{2}$

For $a = 6$: $h = 3\sqrt{3} \approx 5.196$.

Area

From the standard formula $S = \tfrac{1}{2} \times \text{base} \times \text{height}$:

$S = \frac{1}{2} \times a \times \frac{a\sqrt{3}}{2} = \frac{a^2\sqrt{3}}{4}$

For $a = 6$: $S = 9\sqrt{3} \approx 15.588$.

Circumradius and Inradius

Circumradius $R$ (radius of the circle through all three vertices):

$R = \frac{a}{\sqrt{3}} = \frac{a\sqrt{3}}{3}$

Inradius $r$ (radius of the inscribed circle tangent to all three sides):

$r = \frac{a\sqrt{3}}{6} = \frac{h}{3}$

A notable identity: $R = 2r$ for any equilateral triangle. The circumcenter, incenter, centroid, and orthocenter are all the same point (the center of symmetry).

Side $a$	Height $h$	Area $S$	$R$	$r$
1	0.866	0.433	0.577	0.289
6	5.196	15.588	3.464	1.732
10	8.660	43.301	5.774	2.887

The 30-60-90 triangle connection

Every equilateral triangle can be divided into six congruent 30-60-90 triangles by its three medians. The ratios of sides in a 30-60-90 triangle are $1 : \sqrt{3} : 2$, which is why $\sqrt{3}$ appears in all equilateral triangle formulas.

The R = 2r relationship

Place the incenter/circumcenter at the origin. The circumradius $R$ reaches a vertex; the inradius $r$ reaches the midpoint of a side. Each median has length $h = a\sqrt{3}/2$. The centroid divides each median in ratio $2:1$ from the vertex:

• Distance from centroid to vertex: $\tfrac{2}{3} h = \tfrac{a\sqrt{3}}{3} = R$ ✓
• Distance from centroid to midpoint of side: $\tfrac{1}{3} h = \tfrac{a\sqrt{3}}{6} = r$ ✓
• Therefore $R = 2r$ ✓

Frequently Asked Questions (FAQ)

What is the formula for the area of an equilateral triangle?

The area of an equilateral triangle with side a is S = (√3 ÷ 4) × a². For example, a = 6: S = (√3 ÷ 4) × 36 = 9√3 ≈ 15.588. This comes from applying the base × height ÷ 2 formula with height h = a√3 ÷ 2.

How do you find the height of an equilateral triangle?

Drop a perpendicular from any vertex to the opposite side. It bisects that side, creating two congruent 30-60-90 triangles with hypotenuse a and short leg a/2. By the Pythagorean theorem: h² = a² − (a/2)² = 3a²/4, so h = a√3 ÷ 2. For a = 6: h = 3√3 ≈ 5.196.

What is the circumradius of an equilateral triangle?

The circumradius R (radius of the circumscribed circle through all three vertices) is R = a√3 ÷ 3 = a ÷ √3. For a = 6: R = 6 ÷ √3 = 2√3 ≈ 3.464. This follows from the extended Law of Sines: R = a ÷ (2 sin 60°) = a ÷ √3.

Why is the circumradius twice the inradius in an equilateral triangle?

For an equilateral triangle: R = a√3 ÷ 3 and r = a√3 ÷ 6, so R = 2r exactly. Geometrically, the centroid, circumcenter, and incenter all coincide in an equilateral triangle. The centroid divides each median in the ratio 2:1 from the vertex, placing it at two-thirds of the way from vertex to midpoint — which is R from the vertex and r from the side.