Angles in polygons

A polygon is a 2D shape with straight sides. Its interior angles are the angles inside it. From one corner, you can split an n-sided polygon into n − 2 triangles, so:

Sum of interior angles = (n − 2) × 180° (learn this)

Example: a pentagon has angles of 100°, 120°, 95°, 110° and x. Find x.

1. Sum: (5 − 2) × 180 = 540°
2. 100 + 120 + 95 + 110 = 425
3. x = 540 − 425 = 115°

An exterior angle is between one side and the next side extended outwards.

• The exterior angles of any polygon add up to 360°.
• Interior angle + exterior angle = 180° (they make a straight line).

In a regular polygon (all sides and angles equal), each exterior angle is 360° ÷ n.

Example: find the exterior and interior angles of a regular octagon.

1. Exterior angle = 360 ÷ 8 = 45°
2. Interior angle = 180 − 45 = 135°

Check: 8 × 135 = 1080, and (8 − 2) × 180 = 1080.

1. If you have the interior angle, find the exterior angle: 180 − interior.
2. Number of sides = 360 ÷ exterior angle.
3. If the answer isn't a whole number, there is no such regular polygon.

Example: each interior angle of a regular polygon is 156°. How many sides does it have?

1. Exterior angle = 180 − 156 = 24°
2. n = 360 ÷ 24 = 15 sides

Example: can a regular polygon have interior angles of 100°?

1. Exterior angle = 180 − 100 = 80°
2. 360 ÷ 80 = 4.5, which isn't a whole number
3. No