Long Radius of Regular Polygon equals Radius of Circumcircle

Theorem
Let $P$ be a regular polygon.

Let $C$ be the circumcircle of $P$.

The long radius of $P$ is equal to the radius of $C$.

Proof
By definition of circumcircle, $C$ is the circle such that all vertices of $P$ are on $C$.

By definition, the center of $P$ is defined as the center of $C$.

Hence:
 * $OA$ is the radius of $C$

and also:
 * $OA$ is the long radius of $P$.

Hence the result.