Internal angle

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°.


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°.

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