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.
Every polygon, from a triangle to a shape with twenty sides, has corners on the inside called interior angles. If you add them all up, you get the polygon's angle sum. Pick any polygon and count its sides. Call that number n. Now subtract 2 and multiply by 180. That is the whole formula: sum = (n - 2) × 180. Why does it work? Because you can split any polygon into triangles by drawing lines from one corner, and you always get 2 fewer triangles than sides. Each triangle holds 180°, so multiplying the number of triangles by 180 gives the total. Try a hexagon: 6 sides, so (6 - 2) × 180 = 4 × 180 = 720 degrees. If the polygon is regular, all the angles are equal, so divide the sum by n to get one angle. A regular hexagon has 720 ÷ 6 = 120 degrees at each corner.
An interior angle is an angle inside a shape, made where two sides meet. The 3 interior angles of every triangle add up to exactly 180 degrees. It does not matter if the triangle is tall, flat, or tilted. For example, a triangle with angles of 109°, 23°, and 48° works because 109 + 23 + 48 = 180. This fact lets you find a missing angle. If you know two angles of a triangle, add them together, then subtract from 180. Say a triangle has angles of 90° and 35°. Add them to get 125, then do 180 - 125 = 55. The missing angle is 55°. You can use the same idea for bigger shapes too, because any polygon can be cut into triangles.
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?
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