# Equilateral Triangles

An equilateral triangle has three equal angles and sides.

### Perimeter of an equilateral triangle

#### Examples

Calculate the perimeter of an equilateral triangle with a side of 10 cm.

P = 3 · 10 = 30 cm

### Height of an Equilateral Triangle

By applying the Pythagorean theorem for one half of the equilateral triangle, we obtain:

#### Example

Calculate the height of an equilateral triangle with a side of 10 cm.

### Area of an Equilateral Triangle

#### Examples

Calculate the area of an equilateral triangle with a side of 10 cm.

The perimeter of an equilateral triangle measures 0.9 dm and its height is 25.95 cm. Calculate the area of the triangle.

P = 0.9 dm = 90 cm

l = 90 : 3 = 30 cm

A = (30 · 25.95) : 2 = 389.25 cm²

#### Example

Find the area of the following equilateral triangle:

**P = 3 · 10 =** 30 cm

#### Apothem of an Equilateral Triangle

The side of an inscribed equilateral triangle is:

By applying the Pythagorean theorem for the small red triangle, we obtain:

#### Example

Calculate the apothem of an equilateral triangle with a side of 6 cm.

#### Centroid of an Equilateral Triangle

In an equilateral triangle the orthocenter, centroid, circumcenter and incenter coincide.

The center of the circle is the centroid and height coincides with the median. The radius of the circumcircle is equal to two thirds the height.

#### Problems

Calculate the area of an equilateral triangle inscribed in a circle with a radius of 6 cm.

Given an equilateral triangle with a side of 6 cm, find the area of the circular sector determined by the circle circumscribed around the triangle and the radius passing through the vertices.

Calculate the side of an equilateral triangle inscribed in a circle of 10 cm radius.