Two facts do most of the work with polygons. The interior angles of an n-sided polygon add to (n − 2) × 180°, and the exterior angles of any convex polygon add to 360°. For a regular polygon, every exterior angle equals 360° ÷ n, and each interior angle equals 180° minus that.
This is the third lesson in angles and geometric reasoning. It extends the triangle angle sum from the previous lesson.
Why does the interior sum work?
Pick one vertex and draw lines to every other vertex. This splits the polygon into triangles, and there are always n − 2 of them. Each triangle has angles adding to 180°, so the total is (n − 2) × 180°.
| Shape | n | Interior sum |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
How do interior and exterior angles connect?
At any corner, the interior angle and the exterior angle lie along a straight line, so interior + exterior = 180°. For a regular polygon, all exterior angles are equal, so each one is 360° ÷ n.
- Decide whether the question gives an interior or an exterior angle.
- Convert to the exterior angle, because that one links directly to 360°.
- Use exterior angle = 360° ÷ n to find n, or n to find the angle.
- Convert back to the interior angle if the question asks for it.
- Check: does (n − 2) × 180° ÷ n equal the interior angle?
Worked example
A regular polygon has an exterior angle of 24°. Find the number of sides and the size of each interior angle.
Step 1: The exterior angles add to 360°, and they are all equal, so n = 360 ÷ 24 = 15.
Step 2: Each interior angle is 180° − 24° = 156°.
Check: the interior sum is (15 − 2) × 180° = 13 × 180° = 2340°. Dividing by 15 gives 2340 ÷ 15 = 156°, which matches.
The mistake to watch for
A common slip is to use the interior angle in the 360° formula.
Mistaken answer: A regular polygon has an interior angle of 140°, so n = 360 ÷ 140, which is not a whole number.
The student divided 360° by the interior angle. The 360° rule belongs to exterior angles only.
The correction is to convert first: the exterior angle is 180° − 140° = 40°, so n = 360 ÷ 40 = 9. Check: (9 − 2) × 180° = 1260°, and 1260 ÷ 9 = 140°, which matches. If your number of sides comes out as a decimal, that is the sign you used the wrong angle.
Check yourself
Try these, then open each answer.
1. Find the sum of the interior angles of a hexagon.
Show answer
(6 − 2) × 180° = 4 × 180° = 720°.
2. Find each interior angle of a regular octagon.
Show answer
Each exterior angle is 360° ÷ 8 = 45°, so each interior angle is 180° − 45° = 135°. Check: (8 − 2) × 180° = 1080°, and 1080 ÷ 8 = 135°.
3. Four angles of a pentagon are 100°, 110°, 120° and 90°. Find the fifth angle.
Show answer
The interior sum is (5 − 2) × 180° = 540°. The four angles total 100 + 110 + 120 + 90 = 420°, so the fifth is 540° − 420° = 120°.
Where this leads next
Polygon angles lead naturally to circles, where a new set of angle facts applies: apply a circle angle relationship with a reason. The non-calculator working trainer is useful for the multiplication and division here, and the mixed practice set tests interior and exterior angles together.
Some students memorise both formulas but mix them under pressure. That is the kind of slip our teachers trace in online one-to-one Mathematics tuition.