Polygon Interior Angles - Math Open Reference

Definition: The angles on the inside of a polygon formed by each pair of adjacent sides.

(See also Exterior angles of a polygon)

Try this Adjust the polygon below by dragging any orange dot. Click on "make regular" and repeat. Note the behavior of the interior angles and their sum.

The interior angles of a polygon are those angles at each vertex that are on the inside of the polygon. There is one per vertex. So for a polygon with N sides, there are N vertices and N interior angles.

For a regular polygon, by definition, all the interior angles are the same. In the figure above, Click on "make regular" then change the number of sides and resize the polygon by dragging any vertex. Notice that for any given number of sides, all the interior angles are the same.

For an irregular polygon , each angle may be different. Click on "make irregular" and observe what happens when you change the number of sides, and drag a vertex.

The interior angles of any polygon always add up to a constant value, which depends only on the number of sides. For example the interior angles of a pentagon always add up to 540° no matter if it regular or irregular, convex or concave, or what size and shape it is. The sum of the interior angles of a polygon is given by the formula

where
n is the number of sides

So far example:

A square	has 4 sides,	so interior angles add up to 360°
A pentagon	has 5 sides,	so interior angles add up to 540°
A hexagon	has 6 sides,	so interior angles add up to 720°
... etc

For a regular polygon, the total described above is spread evenly among all the interior angles, since they all have the same values. So for example the interior angles of a pentagon always add up to 540°, so in a regular pentagon (5 sides), each one is one fifth of that, or 108°. Or, as a formula, each interior angle of a regular polygon is given by:

where
n is the number of sides

Adjacent angles

Related polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons