Regular Polygons
Definition
A regular polygon is a polygon that is both equilateral (all sides are equal in length) and equiangular (all interior angles are equal in measure).
Examples include the equilateral triangle (3 sides), the square (4 sides), the regular pentagon (5 sides), the regular hexagon (6 sides), and so on.
Key Formulas for a Regular $n$-gon
| Property | Formula |
|---|---|
| Sum of Interior Angles | $S = (n - 2) \times 180°$ |
| Each Interior Angle | $\theta = \dfrac{(n-2) \times 180°}{n}$ |
| Each Exterior Angle | $\phi = \dfrac{360°}{n}$ |
| Number of Diagonals | $d = \dfrac{n(n-3)}{2}$ |
| Area (side $s$) | $A = \dfrac{n s^2}{4} \cot\!\left(\dfrac{\pi}{n}\right)$ |
| Perimeter | $P = n \times s$ |
Important Relationships
Interior + Exterior = 180°
At each vertex, the interior angle and exterior angle are supplementary: $\theta + \phi = 180°$
At each vertex, the interior angle and exterior angle are supplementary: $\theta + \phi = 180°$
Sum of Exterior Angles = 360°
For any convex polygon, the sum of all exterior angles (one at each vertex) is always $360°$.
For any convex polygon, the sum of all exterior angles (one at each vertex) is always $360°$.
Apothem: The distance from the center to the midpoint of a side. For side length $s$:
$$a = \frac{s}{2} \cot\!\left(\frac{\pi}{n}\right)$$
Area can also be written as: $A = \frac{1}{2} \times P \times a$
$$a = \frac{s}{2} \cot\!\left(\frac{\pi}{n}\right)$$
Area can also be written as: $A = \frac{1}{2} \times P \times a$