You already know that a triangle totals 180° and a quadrilateral totals 360°. What about a pentagon, a hexagon, a shape with twenty sides? You do not need a new fact for each one. You need the diagonal trick from the last topic, used again, and it handles every polygon there will ever be.
Cut it into triangles from one corner
Pick one corner of the polygon. Draw a straight line from it to every other corner it is not already joined to. The shape falls apart into triangles, and every one of the polygon corners has been shared out among them with nothing left over. Count the triangles, multiply by 180, and that is the angle sum.
| Shape | Sides | Triangles | Angle sum |
|---|---|---|---|
| Triangle | 3 | 1 | 1 × 180 = 180° |
| Quadrilateral | 4 | 2 | 2 × 180 = 360° |
| Pentagon | 5 | 3 | 3 × 180 = 540° |
| Hexagon | 6 | 4 | 4 × 180 = 720° |
| Octagon | 8 | 6 | 6 × 180 = 1080° |
| Any polygon | n | n - 2 | (n - 2) × 180° |
Stand at the corner you picked. You cannot draw a line to yourself, and you cannot draw one to either of your two neighbours, because you are already joined to both of them by sides. So out of n corners you can reach n minus 3 of them, giving n minus 3 diagonals. Those diagonals cut the shape into n minus 2 pieces, because every cut adds exactly one more piece to what you had.
Regular polygons
A polygon is called regular when all its sides are equal and all its angles are equal. Both conditions are needed. A rhombus has equal sides but unequal angles, so it is not regular. A rectangle has equal angles but unequal sides, so it is not regular either. A square has both, so a square is the regular quadrilateral.
If a polygon is regular, all its angles are the same size, so you can share the total out equally. Each interior angle is (n - 2) × 180 ÷ n. For a regular hexagon that is 720 ÷ 6 = 120°. For a regular pentagon it is 540 ÷ 5 = 108°.
At each corner, the exterior angle is what is left over on the straight line, so interior plus exterior is 180. Now add up all the exterior angles of any polygon at all, regular or not, three sides or thirty. The total is always 360. Imagine walking right round the edge of the shape: at each corner you turn by the exterior angle, and by the time you are back where you started facing the same way, you have turned one full circle.
For a regular polygon, the exterior angle is 360 ÷ n, because the 360 is shared equally among n corners. Then the interior angle is 180 minus that. For a regular hexagon: 360 ÷ 6 = 60, so each interior angle is 180 - 60 = 120°.
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Find the sum of the interior angles of a polygon with 12 sides
Each interior angle of a regular polygon is 150°. How many sides has it?
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Join one corner of a polygon to every corner it is not already joined to. The shape splits into triangles.
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