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.