Overview

Regular polygons are symmetric. Learn the angle and area formulas so you can switch between side length, apothem, and radius quickly.

Core Skills

Use Central Angles

Each central angle is $360^\circ/n$, which helps relate side length to radius.

Use the Apothem

Area is $\frac12 \cdot (\text{perimeter}) \cdot (\text{apothem})$.

Decompose into Triangles

Split the polygon into $n$ congruent isosceles triangles from the center.

Key Ideas

• Central angle: $360^\circ/n$.
• Interior angle: $\frac{(n-2)\cdot 180^\circ}{n}$.
• Area: $\frac{1}{2}\cdot \text{perimeter} \cdot \text{apothem}$.

Worked Example

A regular hexagon has side length $6$. Find its area.

A regular hexagon is 6 equilateral triangles of side $6$. Each has area $\frac{\sqrt{3}}{4}6^2 = 9\sqrt{3}$, so total area is $54\sqrt{3}$.

More Examples

Example 1: Interior Angle

Find the interior angle of a regular decagon.

$\frac{(10-2)\cdot 180^\circ}{10} = 144^\circ$.

Example 2: Radius from Side

In a regular pentagon with side $s$, express the circumradius $R$.

Use $s=2R\sin(36^\circ)$, so $R=s/(2\sin 36^\circ)$.

Example 3: Apothem

A regular hexagon has side $4$. Find its apothem.

Use a $30$-$60$-$90$ triangle: apothem is $2\sqrt{3}$.

Strategy Checklist

• Use central angles to connect side and radius.
• Decompose into congruent triangles for area.
• Keep track of degrees vs radians.

Common Pitfalls

• Using the interior angle formula for central angles.
• Forgetting to convert to radians when using trig area formulas.
• Confusing apothem with radius.

Practice Problems