Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons
You already know that the three angles inside any triangle always add up to 180°. That one fact is the key to every other shape too.
Try a pentagon, which has 5 sides. Draw lines from one corner and it splits into 3 triangles. Three triangles means 3 times 180°, which is 540°. That is the angle sum for any pentagon.
For any polygon with n sides, the number of triangles from one corner is always (n − 2). So the angle sum is (n − 2) × 180°.
In a regular polygon every interior angle is equal, so divide the total by n. Each exterior angle is 180° minus the interior angle, and all the exterior angles of any polygon always add up to 360°.
Two more facts help with regular polygons. In a regular polygon every interior angle is equal, so each one is the total angle sum divided by n. Each exterior angle sits on a straight line with its interior angle, so one exterior angle is 180° minus the interior angle. And if you walk all the way around any polygon, turning one exterior angle at each corner, you make exactly one full turn: the exterior angles of any polygon always add up to 360°, however many sides it has.
Split a polygon into triangles, multiply by 180°, then divide by n for a regular one's interior angle.
You already know every triangle's corners add up to 180°. That one fact unlocks every other shape. A patio is being paved with polygon-shaped stones, starting with a pentagon. Draw lines from one corner of it to the other corners.
How many triangles does the pentagon split into?
This opens up
Nothing builds on it yet.
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.