Definitions

# Internal angle

In geometry, an interior angle (or internal angle) is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon. A simple polygon has exactly one internal angle by vertex.

If every internal angle of a polygon is at most 180 degrees, the polygon is called convex.

In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.

## Interior angle measures of regular polygons

To find the total measure of degrees in a regular polygon, (regular meaning all sides and angles are equal) you must take the number of sides the polygon has, n, subtract 2 from it, then multiply that number by 180°.

Example:

A decagon, a polygon with 10 sides, is a simple shape to figure the total measure of

$\left(n-2\right) times 180^circ !$

# measure in degrees, when n

number of sides

Solution to the decagon:

$\left(10-2\right) times 180^circ =1440^circ. !$

The total measure of the decagon is 1440°.

Divide that number by the number of sides, in this case, 10, to find the measure of each angle.

Each interior angle of a regular decagon is 144°.

It is easier to use measure of an exterior angle. Since every regular polygon can be built from n isosceles triangles, to get the measure of an internal angle simply subtract measure of exterior angle (see below) from 180°

For decagon this gives us:

$180^circ - frac\left\{360^circ\right\}\left\{10\right\} = 180^circ - 36^circ = 144^circ$

For pentagon:

$180^circ - frac\left\{360^circ\right\}\left\{5\right\} = 180^circ - 72^circ = 108^circ$

## Finding the exterior angles on a regular polygon

To find the measure of a regular decagon's exterior angles, divide 360° by the number of sides the polygon has, in this case, 10.

$frac\left\{360^circ\right\}\left\{10\right\} = 36^circ.$

So all the exterior angles in a regular decagon are 36°.