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".
Y9_Polygons_01.gif
Y9_Polygons_02.gif
Y9_Polygons_03.gif

Exterior Angles of Polygons

 

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

e.g. a + b + c + d = 360° Y9_Polygons_04.gif

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.
Y9_Polygons_05.gif

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.

Y9_Polygons_06.gif

Note that the more sides the polygon has, the larger its inside angles are.

 

NewLampBig.jpgRegular Polygons

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