What Are the Angles of a Regular Polygon?

The word polygon is a collective term for all triangles, quadrilaterals, pentagons, and so on.

Theory

Regular Polygons

A figure is regular if all its sides have the same length, and all its angles are the same size.

Examples of regular polygons are:

Formula

The Central Angle

The central angle of a regular polygon is the angle between two lines that go from the center of the polygon to two neighboring vertices.

The central angle s of a regular polygon with n sides can be calculated by using the formula

s = 360 n

Here’s a regular hexagon:

Regular hexagon with central angle

Example 1

You have a regular hexagon. Find the central angle.

You can insert the information you have straight into the formula and find the answer (just look at the figure above):

s = 360 6 = 60

Formula

Interior Angle

The interior angle of a regular polygon is the angle between two neighboring sides.

The interior angle v of a regular polygon with n sides can be found with the equation

v = 180 360 n

This is a regular pentagon:

Regular pentagon with interior angle

Example 2

You have a regular pentagon. Find the interior angle.

You can insert the information you’ve been given straight into the formula and find the answer:

v = 180 360 5 = 108

Formula

The Sum of Angles

The sum of angles a in a polygon with n sides tells you the sum of the interior angles in the polygon. It’s described by this formula:

a = (n 2) 180

Example 3

You have a regular heptagon. Find the sum of its angles.

As a heptagon has seven sides, you can just insert the information you’ve been given into the formula to find the answer:

a = (7 2) 180 = 5 180 = 900

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