Equilateral Triangles

An equilateral triangle has three equal angles and sides.

Equilateral Triangle

Perimeter of an equilateral triangle

Formula for 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

Height of an Equilateral Triangle

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

Equilateral Triangle Operations

Equilateral Triangle Operations

Formula for the Height of an Equilateral Triangle

Example

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

Height of an Equilateral Triangle

Equilateral Triangle Operations

Equilateral Triangle Operations

Area of an Equilateral Triangle

Formula for the Area of an Equilateral Triangle

Examples

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

Equilateral Triangle Operations


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

Height of an Equilateral 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:

Equilateral Triangle

Equilateral Triangle Operations

Equilateral Triangle Operations

P = 3 · 10 = 30 cm

Equilateral Triangle Solution

Apothem of an Equilateral Triangle

Apothem of an Equilateral Triangle

The side of an inscribed equilateral triangle is:

Formula for the Length of a Side of an Equilateral Triangle

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

Equilateral Triangle Operations

Formula for the Apothem of an Equilateral Triangle

Example

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

Apothem of an Equilateral Triangle


Equilateral Triangle Operations


Centroid of an Equilateral Triangle

Orthocenter, Centroid, Circumcenter and Incenter 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.

Formula for the Radius of the Circumcircle

Problems

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

Equilateral Triangle Inscribed in a Circle

Equilateral Triangle Operations

Equilateral Triangle Operations

Equilateral Triangle Operations

Equilateral Triangle Operations

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.

Equilateral Triangle Inscribed in a Circle

Equilateral Triangle Operations

Equilateral Triangle Operations

Equilateral Triangle Solution

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

Equilateral Triangle Inscribed in a Circle

Equilateral Triangle Operations

Equilateral Triangle Operations

Equilateral Triangle Solution