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.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polygon
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polygon