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Central Angle of a Polygon
Definition: The angle subtended at the center of the polygon by one of its sides.
Try this Adjust the number of sides of the polygon below, or drag a vertex to note the central angle of the polygon.

Regular Polygon case

The central angle is the angle made at the center of the polygon by any two adjacent vertices of the polygon. If you were to draw a line from any two adjacent vertices to the center, they would make the central angle. Because the polygon is regular, all central angles are equal. It does not matter which side you choose.

All central angles would add up to 360° (a full circle), so the measure of the central angle is 360 divided by the number of sides. Or, as a formula: where
n  is the number of sides

The measure of the central angle thus depends only on the number of sides. In the figure above, resize the polygon and note that the central angle does not change. The angle depends only on the number of sides, not the size of the polygon. If you change the number of sides, you will see that as the number of sides gets larger, the cenral angle gets smaller.

Irregular Polygon case

Irregular polygons are not considered as having a center, and so have no central angle.
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Other polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons