Polygons • Topic 1 of 3

Interior & Exterior Angles

Drop one diagonal at a time from a single vertex and any convex n-gon falls into (n−2) triangles, so the interior angles must total (n−2)×180°. A triangle gives 180°, a quadrilateral 360°, a pentagon 540°, a hexagon 720° — each step adds 180°. The companion fact is even cleaner: walk once around any convex polygon turning at each corner and you complete one full rotation, so the exterior angles always sum to 360°, no matter how many sides. At every vertex the interior and exterior angle are a linear pair, so they add to 180°. The fast CAT move is to work with exterior angles whenever you can — they are usually smaller, sum to a fixed 360°, and turn "find the number of sides" problems into a one-line division.

✅ Solved examples

1. Find the sum of the interior angles of a decagon (10 sides).
Sum = (n − 2) × 180° = (10 − 2) × 180° = 8 × 180° = 1440°.
2. One interior angle of a regular polygon is 162°. How many sides does it have?
Exterior = 180° − 162° = 18°. Number of sides = 360° / 18° = 20.
3. The interior angles of a pentagon are in the ratio 2 : 3 : 4 : 5 : 6. Find the largest angle.
Sum = (5 − 2) × 180° = 540°. Parts = 2+3+4+5+6 = 20, so one part = 540/20 = 27°. Largest = 6 × 27° = 162°.
4. Four angles of a hexagon are each 100°, and the remaining two are equal. Find each of the equal angles.
Hexagon sum = (6 − 2) × 180° = 720°. Four 100° angles give 400°, leaving 320° for two equal angles ⇒ each = 160°.

✏️ Practice — try these, take hints as needed

1. Find the sum of the interior angles of a 15-sided polygon.
Use (n − 2) × 180°.
n − 2 = 13.
13 × 180°.
2340°
2. Each exterior angle of a regular polygon is 24°. How many sides?
Exterior angles sum to 360°.
n = 360 / each exterior.
360 / 24.
15 sides
3. One interior angle of a regular polygon is 150°. Find the number of sides.
Exterior = 180° − 150°.
Exterior = 30°.
360 / 30.
12 sides
4. The interior angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find the smallest angle.
Sum = 360°.
Parts = 1+2+3+4 = 10.
One part = 36°.
36°
5. Five angles of a hexagon are 110°, 120°, 130°, 140° and 150°. Find the sixth.
Hexagon sum = 720°.
Add the five given: 650°.
720 − 650.
70°

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
Regular Polygons →