SECTION 05
Regular Polygons
A regular polygon satisfies two conditions at the same time: all of its sides are equal in length, and all of its interior angles are equal in measure. If only one condition holds, the polygon is irregular.
One interior angle
Because every interior angle is equal, divide the total sum by the number of angles.
One interior angle = (n − 2) × 180° ÷ n
| Regular polygon | Sides (n) | Calculation | One angle |
|---|---|---|---|
| Equilateral triangle | 3 | 180° ÷ 3 | 60° |
| Square | 4 | 360° ÷ 4 | 90° |
| Regular pentagon | 5 | 540° ÷ 5 | 108° |
| Regular hexagon | 6 | 720° ÷ 6 | 120° |
| Regular octagon | 8 | 1,080° ÷ 8 | 135° |
| Regular decagon | 10 | 1,440° ÷ 10 | 144° |
Worked examples
1. ABCDE is a regular pentagon. Find one interior angle X.
∵ sum of interior angles = 540°
∴ X = 540 ÷ 5
∴ X = 108
2. ABCDEFGH is a regular octagon and JHGI is a square. Find X.
One angle of a regular octagon = 1,080 ÷ 8 = 135°
One angle of a square = 90°
Angles around point H: 135 + 90 + X = 360
∴ X = 135
Try it yourself
Decoration: a regular octagon and a square share the side BC. What is m(∠ABC)?
Answer: 360° − 135° − 90° = 135°.