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5.1 Regular polygons

5.1 Regular polygons

All the angles of a regular polygon are the same size.
All the sides of a regular polygon are the same length.
i°

!is is a regular pentagon.
Each interior angle of a regular polygon is the same size.
i° e°

!e two angles labelled i° are interior angles of this regular pentagon.
You can extend a side of any polygon to make an exterior angle.
!e angle labelled e° is an exterior angle of this pentagon.

Imagine you could walk anticlockwise along The angle is labelled e°,
the sides of the pentagon. so e is a number, without
Start and "nish at P. units. If an angle is labelled

At each corner you turn le# through e°. e, you must include the

A#er "ve turns you have turned 360°, so degrees sign when you

P state the size of the angle.
e = 360 ÷ 5 = 72.
!e exterior angle of the pentagon is 72°.

!e interior angle of the pentagon is 180° − 72° = 108°.

You can use this method for any regular polygon.
Regular polygon, N sides

Exterior angle e = 360 ÷ N Interior angle = 180 − e or 180 − 360 This is a general result.
N
Diagrams in this excerise are not drawn accurately.

Worked example 5.1

The interior angle of a regular polygon is 140°.
How many sides does the polygon have?
140° 180° – 140°

The exterior angle is 180° − 140° = 40°.
The number of exterior angles is 360° ÷ 40° = 9. Number of angles × 40° = 360°
The regular polygon has nine sides. Nine exterior angles, nine sides

) Exercise 5.1

1 a Write down the usual name for:
i a regular quadrilateral ii a regular triangle.
b Find the interior and exterior angles of:
i a regular quadrilateral ii a regular triangle.

2 Work out the following angles, giving reasons.
a the exterior angle of a regular hexagon
b the interior angle of a regular hexagon

42 5 Shapes