Apothem of Regular Polygon equals Radius of Incircle

Theorem

Let $P$ be a regular polygon.

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

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

Proof

By definition of incircle, $C$ is the circle such that all sides of $P$ are tangent to $C$.

From Regular Polygon can be Circumscribed around Circle, it is established that the center of $P$ and the center of $C$ are the same point.

Hence: the radius of $C$ is the same thing as: the perpendicular distance from 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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): apothem
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): apothem