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Equilateral Triangle Calculator

Calculate equilateral triangle area, height, perimeter, and inradius from the side length. All angles are 60\u00B0.

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Yield Sign (Side 75 cm)

A standard yield road sign is an equilateral triangle with sides of approximately 75 cm.

Key values: side = 75 cm · area ≈ 2,437 cm² · height ≈ 64.95 cm

Steel Frame Triangle (Side 2 m)

A 2 m equilateral triangular steel frame for a structural brace.

Key values: side = 2 m · area ≈ 1.732 m² · perimeter = 6 m

Documentation

Equilateral Triangle Properties

An equilateral triangle has all three sides equal and all three angles equal to 60°. It is the simplest regular polygon and the only triangle where every center point (centroid, circumcenter, incenter, orthocenter) coincides.

Every equilateral triangle is determined by a single measurement: the side length $s$ . All other properties follow from it.

Complete Formula Set

Given side length $s$ :

Height (altitude)

h = \frac{s\sqrt{3}}{2}

Derived from the 30-60-90 triangle formed by dropping an altitude.

Area

A = \frac{s^2\sqrt{3}}{4}

From $A = \frac{1}{2} \times s \times h = \frac{1}{2} \times s \times \frac{s\sqrt{3}}{2}$ .

Perimeter

P = 3s

Circumradius (circumscribed circle)

R = \frac{s}{\sqrt{3}} = \frac{s\sqrt{3}}{3}

Inradius (inscribed circle)

r = \frac{s}{2\sqrt{3}} = \frac{s\sqrt{3}}{6}

The inradius is exactly half the circumradius: $r = R/2$ .

Why $\sqrt{3}$ Appears Everywhere

Drop an altitude from any vertex to the opposite side. This bisects the base, creating two 30-60-90 right triangles with legs $s/2$ and $h$ , and hypotenuse $s$ :

h^2 + \left(\frac{s}{2}\right)^2 = s^2 \implies h = \frac{s\sqrt{3}}{2}

The factor $\sqrt{3}/2 \approx 0.866$ is the ratio of height to side. Every equilateral triangle formula derives from this single geometric fact.

Tessellation and Geometry

The equilateral triangle is one of only three regular polygons that tile the plane (along with the square and regular hexagon). Six equilateral triangles meet at each vertex because $6 \times 60° = 360°$ .

Connection to hexagons: A regular hexagon is made of exactly 6 equilateral triangles. The area of a regular hexagon with side $s$ is therefore $6 \times \frac{s^2\sqrt{3}}{4} = \frac{3s^2\sqrt{3}}{2}$ .

Frequently Asked Questions

What defines an equilateral triangle?

An equilateral triangle has all three sides equal in length and all three interior angles equal to $60°$ . It is the simplest regular polygon, and every center point (centroid, circumcenter, incenter, orthocenter) coincides at the same location.

How do I calculate the area of an equilateral triangle?

Given side length $s$ , the area is $A = \frac{s^2\sqrt{3}}{4}$ . This is derived from the base-height formula $A = \frac{1}{2} \times s \times \frac{s\sqrt{3}}{2}$ , where the height comes from the 30-60-90 triangle formed by dropping an altitude.

What is the height of an equilateral triangle?

The height (altitude) of an equilateral triangle with side $s$ is $h = \frac{s\sqrt{3}}{2}$ . This is derived from the Pythagorean theorem applied to the 30-60-90 right triangle formed when an altitude bisects the base.

What is the relationship between circumradius and inradius?

In an equilateral triangle, the inradius is exactly half the circumradius: $r = R/2$ . The circumradius is $R = \frac{s\sqrt{3}}{3}$ and the inradius is $r = \frac{s\sqrt{3}}{6}$ , where $s$ is the side length.

Why do equilateral triangles tile the plane?

Equilateral triangles tile the plane because six of them fit perfectly around a single vertex: $6 \times 60° = 360°$ . They are one of only three regular polygons that can tessellate the plane, along with squares and regular hexagons.

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