Angles in polygons

Why do bees build their honeycomb out of hexagons? One reason is angles. Each corner of a regular hexagon is 120°, and three corners of 120° fit around a point to make exactly 360°, a full turn. So hexagons fit together with no gaps. Regular pentagons cannot do this: their corners are 108°, and no number of them makes exactly 360°. Numbers like 120° and 108° are easy to work out. They come from one fact you may already know: the angles of a triangle add up to 180°.

Inside angles and triangles

The angles at the corners inside a polygon are called its interior angles, or inside angles. To find what they add up to, cut the polygon into triangles. Choose one corner. From it, draw a straight line to every corner that is not next to it. These lines are diagonals, and they cut the shape into triangles.

A pentagon is cut into 3 triangles this way, and a hexagon into 4. There are always 2 fewer triangles than sides. The two sides that touch your chosen corner do not get a triangle of their own. Every other side has one.

The angles of all the triangles together make exactly the inside angles of the polygon. Each triangle gives 180°.

Four-sided shapes

A quadrilateral is any shape with four sides: a square, a rectangle, a kite, or a shape with four sides of different lengths. One diagonal cuts it into 2 triangles, so its four inside angles always add up to 2 × 180° = 360°.

Each angle of a regular polygon

In a regular polygon all the sides are equal and all the inside angles are equal. So you find the total first, and then share it equally between the corners.

1. Find the sum of the inside angles: (n − 2) × 180°.
2. Divide by the number of corners, n.

Exterior angles

Make one side of a polygon longer, past the corner, with a ruler. The angle between this longer line and the next side is called an exterior angle, or outside angle. The exterior angle and the inside angle at the same corner sit on a straight line, so together they make 180°.

Now imagine walking all the way round the edge of a polygon. At every corner you turn by the exterior angle. When you are back at the start, facing the same way as before, you have made exactly one full turn. So the exterior angles add up to 360°. This is true for every polygon, whatever its number of sides.

You can also go backwards. If you know one exterior angle of a regular polygon, you can find how many sides it has. Ask how many of those angles fit into 360°.