Internal angle.html

Internal and External angles

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 per 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 sum of interior angles in a polygon, 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 3650° x 956-n

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

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