What you'll learn
This topic covers the interior and exterior angles of polygons. In this guide you will learn how to find the sum of interior angles, the size of each angle in a regular polygon, the exterior angle rule, and how to work out the number of sides from an angle. These are core geometry results used in many problems.
Key terms and definitions
Polygon — a closed shape with straight sides.
Regular polygon — a polygon with all sides and all angles equal.
Interior angle — an angle inside the polygon at a vertex.
Exterior angle — the angle between a side and the extension of the next side.
Sum of interior angles — the total of all the interior angles.
Core concepts
Sum of interior angles
The interior angles of a polygon add up to (n − 2) × 180°, where n is the number of sides. This works because a polygon splits into (n − 2) triangles. A pentagon (n = 5) sums to 3 × 180° = 540°.
Interior angle of a regular polygon
In a regular polygon, all interior angles are equal, so each is the sum divided by n: (n − 2) × 180° ÷ n. For a regular hexagon: 720° ÷ 6 = 120°.
Exterior angles sum to 360°
The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle is 360° ÷ n. This is often the quickest route into a problem.
Interior and exterior together
At each vertex, the interior and exterior angles lie on a straight line, so they add up to 180°. Knowing one gives the other instantly.
Finding the number of sides
If you know an exterior angle, the number of sides is 360° ÷ exterior angle. If you know an interior angle of a regular polygon, find the exterior angle (180° − interior) first, then divide into 360°.
Worked examples
Example 1: Interior sum
Find the sum of the interior angles of an octagon.
(8 − 2) × 180 = 1080°.
Example 2: Regular interior angle
Find each interior angle of a regular pentagon.
(5 − 2) × 180 ÷ 5 = 540 ÷ 5 = 108°.
Example 3: Number of sides
A regular polygon has exterior angles of 24°. How many sides?
360 ÷ 24 = 15 sides.
Common mistakes and how to avoid them
Wrong interior sum formula. Use (n − 2) × 180°, not n × 180°.
Mixing interior and exterior. They add to 180° at each vertex.
Forgetting exteriors sum to 360°. This holds for any polygon.
Dividing by the wrong number. For a regular polygon divide the sum by n.
Using regular rules on irregular shapes. Equal-angle rules need a regular polygon.
Exam technique for Polygon Angles
Use (n − 2) × 180° for the interior sum.
Divide by n for each angle of a regular polygon.
Use 360° ÷ n for each exterior angle.
Remember interior + exterior = 180° at a vertex.
Use 360° ÷ exterior angle to find the number of sides.
Quick revision summary
The interior angles of a polygon sum to (n − 2) × 180°, because the polygon splits into (n − 2) triangles (a pentagon sums to 540°). In a regular polygon, each interior angle is the sum divided by n (a regular hexagon: 720° ÷ 6 = 120°). The exterior angles of any polygon sum to 360°, so each exterior angle of a regular polygon is 360° ÷ n. At each vertex the interior and exterior angles add to 180°, so one gives the other. To find the number of sides, divide 360° by the exterior angle (finding the exterior from 180° − interior if needed). The common errors are using the wrong interior-sum formula, mixing interior and exterior angles, forgetting exteriors total 360°, and applying regular rules to irregular shapes. Use (n − 2) × 180° for the sum, divide by n for regular angles, use 360° for exteriors, and 360° ÷ exterior for the number of sides.