Interior and exterior angles of polygons

Learning outcome

Calculate polygon angle sums, identify the correct exterior turn, and determine when an angle can reveal the number of sides.

Concepts and assumptions

Use simple, strictly convex polygons with n ≥ 3 sides: sides do not cross, each interior angle lies between 0° and 180°, and consecutive sides are not collinear.

Drawing diagonals from one vertex partitions a convex polygon into n − 2 triangles. Therefore:

Interior-angle sum = (n − 2) × 180°.

This sum does not require equal sides or equal angles. An individual interior angle cannot generally be found by dividing the sum by n.

For exterior angles, walk once around the boundary in one fixed direction. Extend each incoming side forward and measure the non-reflex turn toward the outgoing side. Count one turn per vertex, consistently clockwise or consistently anticlockwise. These turn magnitudes total 360°.

Interior angle + corresponding exterior turn = 180°.

Here, “exterior angle” means that turn, not a reflex angle outside the polygon. Do not count both exterior angles at a vertex. Concave and self-crossing paths are outside this convention.

A regular polygon has equal sides and equal angles. Equal-angle calculations also apply to a convex equiangular polygon even without equal sides:

Each exterior turn = 360° ÷ n.

Each interior angle = 180° − 360° ÷ n.

Consequently, n = 360° ÷ exterior turn requires equal turns, and the result must be an integer at least 3.

Worked examples

Example 1 — Sum without regularity. A convex polygon’s interior angles total 1,440°. Then (n − 2) × 180 = 1,440, so n − 2 = 8 and n = 10. It has ten sides, but its individual angles need not equal 144°.

Example 2 — Interior-to-exterior ratio. In a regular polygon, an interior angle and its exterior turn are in the ratio 7:2. Write them as 7k° and 2k°. Since 9k = 180, k = 20. The angles are 140° and 40°; therefore n = 360 ÷ 40 = 9.

Example 3 — An irregular pentagon. Convex ABCDE has interior angles 110°, 125°, 100° and 130° at A, B, C and D. Its total is (5 − 2) × 180° = 540°. Hence ∠E = 540° − 465° = 75°, and E’s exterior turn is 180° − 75° = 105°. No regularity is assumed.

Common mistakes

Do not use n × 180° for the interior sum. Do not confuse a total of 360° with equal exterior angles. Never round a noninteger side count into a valid regular polygon.

Practice questions

1. Six interior angles of a convex heptagon are 120°, 130°, 140°, 110°, 150° and 125°. Find the seventh.
2. A regular convex polygon has an exterior turn of 24°. Find its side count and each interior angle.
3. Each interior angle of a regular convex polygon is 165°. Find its side count.
4. Can a regular convex polygon have an exterior turn of exactly 32°? Explain.

Worked solutions

1. Seven sides give a total of (7 − 2) × 180° = 900°. The six given angles total 775°, so the seventh is 900° − 775° = 125°.
2. Equal turns give n = 360 ÷ 24 = 15. Each interior angle is 180° − 24° = 156°.
3. Each exterior turn is 180° − 165° = 15°. Thus n = 360 ÷ 15 = 24.
4. Equal turns would require n = 360 ÷ 32 = 11.25. A polygon cannot have a fractional number of sides, so no such regular convex polygon exists.