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

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

measure in degrees, when n

number of sides

Solution to the decagon:

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

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:

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

For pentagon:

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

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.

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

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

External links

• Internal angles of a triangle and External angles of a triangle With interactive animation
• Angle definition pages with interactive applets that are also useful in a classroom setting. Math Open Reference