Formulas for the Area of a Regular Polygon, Its Side, and the Radius of the Inscribed Circle
Key Idea: A regular polygon is a polygon with all sides equal and all interior angles equal. Every regular polygon has a unique inscribed circle (incircle) and a unique circumscribed circle (circumcircle). The relationships between the number of sides, the side length, the inradius, and the area are governed by elegant trigonometric formulas.
Definitions
| Symbol | Meaning |
|---|---|
| $n$ | Number of sides of the regular polygon |
| $a$ | Length of one side |
| $r$ | Radius of the inscribed circle (apothem) — distance from center to the midpoint of a side |
| $R$ | Radius of the circumscribed circle — distance from center to a vertex |
| $S$ | Area of the polygon |
| $P$ | Perimeter of the polygon, $P = n \cdot a$ |
Interactive Diagram
Interactive visualization — available in the full lesson