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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.

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.

## Radius 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
sin  is the sine function calculated in degrees
(see Trigonometry Overview)

## Radius 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
cos  is the cosine function calculated in degrees
(see Trigonometry Overview)