Angles & Polygons

Any polygon splits into triangles. Draw every diagonal from one corner and an n-sided shape becomes n − 2 triangles. Each triangle holds 180°, which is exactly where (n − 2) × 180 comes from. You never need to memorise it, you can rebuild it.

The exterior angles always add to 360°, whatever n is. Walk right round the outside of the shape and you turn through one full circle by the time you are back where you started. More sides just means smaller turns.

Interior and exterior are supplementary. They sit on a straight line, so they add to 180°. If you know one you know the other, which is usually the fastest route through an exam question.

• Try this: Set n = 3. What is the interior sum, and does it match what you know about a triangle?
• Try this: Watch the exterior-angle total as you go from 3 sides to 12. Does it ever change?
• Try this: At what n does each interior angle pass 120°, and which shape is that?