# Equilateral Triangle Calculator

Calculate equilateral triangle area, perimeter, height, inradius, and circumradius from side length. Free online calculator with instant results.

## What this calculates

Calculate every property of an equilateral triangle from its side length. Find the area, perimeter, height, inradius (inscribed circle), and circumradius (circumscribed circle) instantly.

## Inputs

- **Side Length** — min 0 — The length of each side of the equilateral triangle.

## Outputs

- **Area** — The area of the equilateral triangle.
- **Perimeter** — The total length of all three sides.
- **Height** — The perpendicular height from base to vertex.
- **Inradius** — The radius of the inscribed circle.
- **Circumradius** — The radius of the circumscribed circle.

## Details

An equilateral triangle has three equal sides and three equal angles of 60 degrees each. It is the most symmetric triangle and has several elegant mathematical properties.

Equilateral Triangle Formulas (side length = s):

- Area: A = (sqrt(3) / 4) x s^2

- Perimeter: P = 3s

- Height: h = (sqrt(3) / 2) x s

- Inradius: r = s / (2 x sqrt(3)) = s x sqrt(3) / 6

- Circumradius: R = s / sqrt(3) = s x sqrt(3) / 3

The inradius is exactly half the circumradius (r = R/2) for equilateral triangles, and the centroid, circumcenter, incenter, and orthocenter all coincide at the same point. The height creates two 30-60-90 special right triangles.

Equilateral triangles are used in structural engineering (trusses, geodesic domes), road signs (yield signs), tessellations, and the construction of other geometric shapes. Six equilateral triangles form a regular hexagon.

## Frequently Asked Questions

**Q: What is an equilateral triangle?**

A: An equilateral triangle is a triangle with all three sides equal in length and all three interior angles equal to 60 degrees. It is the only triangle that is also a regular polygon. It has three lines of symmetry and rotational symmetry of order 3.

**Q: How do I find the height of an equilateral triangle?**

A: The height of an equilateral triangle with side s is h = (sqrt(3) / 2) x s, which is approximately 0.866 x s. This comes from the Pythagorean theorem applied to the 30-60-90 right triangle formed by dropping an altitude: h = sqrt(s^2 - (s/2)^2) = sqrt(3s^2/4).

**Q: What is the inradius of an equilateral triangle?**

A: The inradius is the radius of the largest circle that fits inside the triangle (tangent to all three sides). For an equilateral triangle with side s, the inradius is r = s / (2 x sqrt(3)), which equals approximately 0.2887 x s. It equals one-third of the height.

**Q: What is the relationship between inradius and circumradius?**

A: For an equilateral triangle, the circumradius R is exactly twice the inradius r: R = 2r. The circumradius is s / sqrt(3) and the inradius is s / (2 x sqrt(3)). The circumcenter and incenter coincide at the centroid of the triangle.

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Source: https://vastcalc.com/calculators/math/equilateral-triangle
Category: Math
Last updated: 2026-04-21