How to find the height of an equilateral triangle using pythagorean theorem

by Bernat Requena Serra · Published 20 September, 2018 · Updated 14 January, 2021

The equilateral triangle is the regular polygon simplest. Its three sides are equal. Therefore, their angles are also the three equal.

Since all angles are equal and the angles are the sum of 180º, their three interior angles are 60º (180º/3=60º).

Note: Why do they add their angles 180º?

Altitude of an Equilateral Triangle

The altitude (h) of the equilateral triangle (or the height) can be calculated from Pythagorean theorem. The sides a, a/2 and h form a right triangle. The sides a/2 and h are the legs and a the hypotenuse.

Applying the Pythagorean theorem:

And we obtain that the height (h) of equilateral triangle is:

Another procedure to calculate its height would be from trigonometric ratios.

With respect to the angle of 60º, the ratio between altitude h and the hypotenuse of triangle a is equal to sine of 60º. Therefore:

Area of an Equilateral Triangle

An equilateral triangle has three equal sides and angles. As in any type of triangle, its area is equal to half of the product of its base and height. So if the altitude of an equilateral triangle is:

The area it will be defined by the following formula:

Perimeter of an equilateral triangle

The equilateral triangle has all three sides equal, so its perimeter will be three times the length of one of its sides (a).

Download this calculator to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar         or   Triangle-total.exe      

Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.

Resolved Exercises

Exercise of the Equilateral Triangle Area

Find the area of an equilateral triangle in which its three equal sides have the length a=5 cm.

What is its area?

Applying the above formula:

The area is 10.83 cm2.

Exercise of the Equilateral Triangle Perimeter

Being a equilateral triangle with all sides equal in length a=5 cm.

What is its perimeter?

Applying the above formula:

We obtain that the perimeter of this equilateral triangle is 15 cm.