Introduction

As you can see in the

This regularity of form makes regular polygons useful in a variety of situations, as illustrated above.

As you can see in the

*Looking at Polygons*unit, regular polygons have equal-length sides and interior angles of equal size.This regularity of form makes regular polygons useful in a variety of situations, as illustrated above.

Exterior angles

Like any

**convex**

**polygon**

**regular polygon***n*-sided polygon has

*n*exterior angles, each

**angle**Interior angles

All interior angles in a regular polygon are equal. Any

**interior angle**

**straight line***y = ax + c*. It has length and position, but no breadth and is therefore one-dimensional.straight line.

We also know that each exterior angle of an

*n*-sided polygon is equal to . So:

This

**equation**If the interior angles of a regular polygon are 150, how many sides does it have?

It has 12 sides. |

Circumscribing regular polygons

Every regular polygon can 'fit' within a circle so that each of its vertices touches the circle edge.
To see why this works, try the following exercise.

Click on the figure below to interact with the model.

- Click on the
*Paste new triangle*button to paste a copy of the triangle onto the model (it will appear over the first triangle). - Click on the new triangle to select it.
- Use the handle that appears to rotate the triangle until it is next to, but not overlapping, the previous triangle.
- Now repeat this pasting and rotating process, until there are eight triangles in the model.

Do you notice anything about the eight lengths radiating from

**point**- A point has no properties except position. It is an object with zero dimensions.
- Points in the
*x*-*y*plane can be specified using*x*and*y*coordinates.

*C*? Because the eight triangles are isosceles and congruent, each of these lengths is the same, 3 cm.

The fact that any regular

*n*-sided polygon can be divided into

*n*congruent isosceles triangles means that the angle formed at the centre within each

**triangle**Constructing regular polygons

The fact that all regular polygons can be circumscribed makes constructing them very easy. We don't need to construct *n*separate triangles to form an

*n*-sided polygon, we need only do the following:

- Choose the number of sides your regular polygon is to have.
- Work out the angle,
*a*, formed in the centre of each congruent triangle. - Using your compass draw a circle, centre
*O*. - Draw a line, using a ruler, from
*O*to any point on the circumference of the circle. - Using a protractor and a ruler, draw a line, at an angle
*a*to the first line, from*O*to the circumference of the circle. - Set your compasses to the distance between the two points at which the first and second lines meet the circle. Starting at
one of these points, use your compass to mark off points at this same distance around the circle. These points will become
the vertices of your polygon.
- Using a straight edge or ruler, join each point to the next point on the circumference of the circle until your polygon is
complete.

Enter a value between 3 and 20 for

*n*in Fig.7 below and advance through the stages in the construction process to see the steps in drawing a regular polygon.

So constructing a hexagon is extremely easy. No calculations need to be made: if you want to construct a hexagon with side length

*d*cm, you:

- Draw a circle with radius
*d*cm. - Keep the compass set as it is and mark off 6 points on the circle, each one
*d*cm from the last. - Join each point to the next one on the circle with a straight line.

Here's a diagram of how your page should look if you draw a hexagon the simple way.

Summary

All exterior angles in a regular polygon are equal. Each is equal to , where

All interior angles in a regular polygon are equal. Each is equal to , where

Every regular polygon has a circumscribing circle.

All exterior angles in a regular polygon are equal. Each is equal to , where

*n*is the number of sides.All interior angles in a regular polygon are equal. Each is equal to , where

*n*is the number of sides.Every regular polygon has a circumscribing circle.

Well done!

Try again!