Regular Polygons

Regular Polygons

GCSE(F), GCSE(H),

A regular polygon has sides that are all the same length and angles that are all the same size.

As with irregular polygons, the sum of the exterior angles is 360º. The exterior angles are all the same size; an individual angle can be calculated as:

Exterior angle = `frac(360)(text(number of sides)`

Similarly, an individual interior angle can be calculated from the number of sides:

The sum of the interior angles

= (number of sides - 2) x 180

and to calculate an individual angle:

= `frac(text(sum of interior angles))(text(number of sides))`

A line drawn across a polygon to create a triangle (using what are known as Vertically Opposite Angles) creates an Isosceles triangle. Angle CAB is equal to angle ACB.

Examples

1. A regular polygon has an interior angle of 150º. How many sides does the polygon have?

Answer: 20

Calculate the exterior angle: 180 - 150 = 30º
Divide 360 by the exterior angle: 360 ÷ 30 = 20

2. What is the size of angle A?

Answer: 160º

It is a 9-sided shape. Work out the exterior angle: 360 ÷ 9 = 40º

The interior angle at B is 180 - 40 = 140º

The shape ABC is an Isosceles triangle:

the other angles of the triangle are (180 - 140) ÷ 2 = 20º

Angle A is a straight line: 180 - 20 = 160º