# Equilateral triangle

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 cm ^{2}**.

### 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**.