Circumscribed Circle of a Regular Polygon
What is a Circumscribed Circle?
A circumscribed circle (also called a circumcircle) of a regular polygon is the circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter, and its radius is called the circumradius, denoted by $R$.
Key Insight: Every regular polygon has exactly one circumscribed circle. This is because all vertices of a regular polygon are equidistant from its center — and that common distance is the circumradius $R$.
Formal Definition
Given a regular polygon with $n$ sides, each of length $a$:
- Central Angle: The angle subtended at the center by one side is $\theta = \dfrac{360°}{n} = \dfrac{2\pi}{n}$
- Circumradius Formula: $$R = \frac{a}{2 \sin\left(\dfrac{\pi}{n}\right)}$$
- Apothem (Inradius): The distance from the center to the midpoint of a side: $$r = R \cos\left(\frac{\pi}{n}\right) = \frac{a}{2 \tan\left(\dfrac{\pi}{n}\right)}$$