Worked Example 1
Find the sum of the interior angles of a polygon with 12 sides
MethodJoining one corner to the others cuts the shape into n - 2 triangles, and each triangle gives 180°.
Workingn = 12, so triangles = 12 - 2 = 10. Angle sum = 10 × 180 = 1800°.
NoteThe question did not say the polygon was regular, and it did not need to. The angle sum is the same whether the shape is neat or badly squashed.
Worked Example 2
Each interior angle of a regular polygon is 150°. How many sides has it?
The slow waySet up (n - 2) × 180 ÷ n = 150 and solve for n. It works, but it needs algebra you may not want to do under time pressure.
The fast wayGo through the exterior angle instead. Interior + exterior = 180, so the exterior angle is 180 - 150 = 30°. The exterior angles total 360, so n = 360 ÷ 30 = 12 sides.
CheckA 12 sided polygon has angle sum (12 - 2) × 180 = 1800, and 1800 ÷ 12 = 150. Correct.