Height of an Equilateral Triangle

Height of an Equilateral Triangle

How to find the height of an equilateral triangle: definition, formula, proof, examples, and their solutions.

Equilateral Triangle

An equilateral triangle is a triangle whose lengths of the sides are all equal. Its interior angles are all congruent: 60 degrees.

An equilateral triangle is a triangle
whose sides are all congruent.
(See its name: 'equi' + 'lateral'.)

Its interior angles are also all congruent: 60º.

Formula

If the length of an equilateral triangle's side is a, then the height of the equilateral triangle is ((square root 3)/2)*a.

If the equilateral triangle's side is a,
then the height of the equilateral triangle is
(√3/2)⋅a.

Proof

See the left half of the equilateral triangle. It's a right triangle. And it's non-right interior angle is 60 degrees. So this is a 30-60-90 triangle. Draw a (1, square root 3, 2) triangle next to the equilateral triangle. Then set a proportion between the triangles.

See the left half of the equilateral triangle (blue).
The measures of its interior angles are 60 and 90.

So this is a 30-60-90 triangle.

Draw a 30-60-90 triangle
next to the equilateral triangle.
(with its sides → 1 : √3: 2)

The right triangles are similar.

So set a proportion
between the right triangles' corresponding sides:
h / √3 = a / 2.

Similarity of sides in triangles

Example 1

Find the value of x. The length of the equilateral triangle's side: 8. The height of the equilateral triangle: x.

The equilateral triangle's side: 8

Height: h = (√3/2)⋅8

Example 2

Find the value of x. The length of the equilateral triangle's side: x. The height of the equilateral triangle: 12.

The equilateral triangle's side: x

Height: 12 = (√3/2)⋅x

Rationalizing the denominator