# A regular polygon has interior angles that are 5 times larger than each of its exterior angles. How many sides does the polygon have?

Nov 30, 2015

$12$

#### Explanation:

The exterior angles of a dodecagon are ${30}^{\circ}$, so they sum to $12 \times {30}^{\circ} = {360}^{\circ}$

The interior angles are ${150}^{\circ} = {180}^{\circ} - {30}^{\circ} = 5 \times {30}^{\circ}$

May 1, 2016

We need to be able to calculate this, without having to consider the size of the exterior and interior angles of all the different polygons.

#### Explanation:

Let the size of an exterior angle be x°
The size of the interior angle is therefore 5x°

An exterior and interior angle are supplementary angles.

x° + 5x° = 180° rArr 6x = 180°

x = 30° This is size of each exterior angle ($e x t \angle$)

The sum of the exterior angles is 360°

Number of sides (or angles) = $\frac{360}{e x t \angle}$
$\frac{360}{30} = 12$ sides