## Polygons

A polygon is a shape with three or more straight sides.

### Types of Polygon

 Regular Polygons Convex Polygons Concave Polygons A regular polygon has all sides and all angles equal. A convex polygon has all sides pointing "outwards". A concave polygon has at least one pair of sides pointing "inwards".   ### Exterior Angles of Polygons

The exterior (outside) angles of all polygons add up to 360°.

e.g. a + b + c + d = 360° ### Interior Angles of Polygons

The interior (inside) angles of a polygon have different sums depending on the number of sides in the polygon.

The sum of the interior angles can be investigating how many triangles can fit into the polygon.

 e.g. The diagram shows a pentagon. Three triangles can be drawn inside the pentagon. Each triangle has interior angles with a sum of 180° So the total inside angles of the pentagon must be 3 × 180 = 540° (5 sides fits 3 triangles − n sides would fit (n − 2) triangles) In general: The sum of the interior angles of a polygon with n sides = (n − 2) × 180

A spreadsheet can be used to look at the sum of interior angles of other polygons. Note that the more sides the polygon has, the larger its inside angles are.

The table summarises the names and angles of the more common regular polygons .

 Number of sides Name Size of each exterior angle Size of each interior angle 3 Equilateral triangle 360⁄3 = 120° 180 − 120 = 60° 4 Square 360⁄4 = 90° 180 − 90 = 90° 5 Pentagon 360⁄5 = 72° 180 − 72 = 108° 6 Hexagon 360⁄6 = 60° 180 − 60 = 120° 8 Octagon 360⁄8 = 45° 180 − 45 = 135° 10 Decagon 360⁄10 = 36° 180 − 36 = 144° n n − gon 360⁄n 180 − 360⁄n