What Are the Angles of a Regular Polygon?

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

Theory

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 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 = \frac{360^{\circ}}{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 = \frac{360^{\circ}}{6} = 60^{\circ}

Formula

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^{\circ} - \frac{360^{\circ}}{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^{\circ} - \frac{360^{\circ}}{5} = 108^{\circ}

Formula

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) \cdot 180^{\circ}

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) \cdot 180^{\circ} = 5 \cdot 180^{\circ} = 900^{\circ}

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