Radius of a Regular Polygon - Math Open Reference

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Radius of a regular polygon (also Circumradius)

Definition: The distance from the center of a regular polygon to any vertex .

Try this Adjust the polygon below by dragging any orange dot, or alter the number of sides. Note the behavior of the polygon's radius.

(If there is no image below, see support page.)

The radius of a regular polygon is the distance from the center to any vertex. It will be the same for any vertex. The radius is also the radius of the polygon's circumcircle, which is the circle that passes through every vertex. In this role, it is sometimes called the circumradius.

Irregular polygons are not usually thought of as having a center or radius.

Finding the radius

The formulas below are for the radius of a regular polygon. Use the formula that uses the facts you are given to start.

Given the length of a side:

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the radius is given by the formula:

where
S is the length of any side
N is the number of sides
π is PI, approximately 3.142
SIN is the sine function calculated in radians (see Trigonometry Overview)

Given the apothem (inradius):

If you know the apothem (or inradius) (distance from the center to the midpoint of a side):

where
A is the apothem (inradius)
N is the number of sides
π is PI, approximately 3.142
COS is the cosine function calculated in radians (see Trigonometry Overview)

Related polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons