# Internal angle

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Internal angle

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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

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

= measure in degrees, when "n" = number of sides

Solution to the decagon:

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

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°.