Drawing Regular Polygons with a Protractor

16 min

Let's begin

Mutesi is designing a badge for the school's environment club: a regular pentagon with a tree in the middle. She draws five sides by eye, and the last side comes out too short. Her friend laughs: "That's not a pentagon, that's a sad house!"

To draw a regular polygon properly you need two things: equal sides, and the right angle at every corner. Here are two reliable ways to do it.

Method 1: side and interior angle

Use this method when you are told the length of a side. You need the interior angle of the polygon:

Regular polygon	Equilateral triangle	Square	Pentagon	Hexagon	Octagon
Interior angle	60°	90°	108°	120°	135°

Worked example 1

a regular pentagon with sides of 5 cm.

Draw side AB 5 cm long.
At A, measure 108° and draw a line. At B, measure 108° on the same side and draw a line.
Open your compasses to 5 cm. Put the point on A and mark E on the line from A. Put the point on B and mark C on the line from B.
At E, measure 108° and draw a line. Mark D on it, 5 cm from E.
Join C to D. Check that CD is 5 cm and that the angles at C and D are 108°.

Method 2: the circle and the central angle

Use this method when the polygon must fit inside a circle. You need the central angle, 360° ÷ n.

Draw a circle and mark its centre O.
Draw a radius OA.
Measure the central angle from OA and draw the next radius. Where it meets the circle, mark B.
Keep going round the circle until you are back at A.
Join the points on the circle in order.

Worked example 2

a regular hexagon in a circle of radius 4 cm.

Central angle = 360° ÷ 6 = 60°.
Draw radii at 0°, 60°, 120°, 180°, 240° and 300° from OA.
Join the six points. Measure a side: it is 4 cm, the same as the radius. This is true for every regular hexagon.

Worked example 3

a regular pentagon in a circle. The central angle is 360° ÷ 5 = 72°, so draw radii 72° apart.

Check your drawing

Measure every side and every angle. They should all match. Small errors grow as you go round, so use a sharp pencil and measure from the exact point each time.

Tip

Know the side? Use side-and-angle. Know the circle? Use the centre. Either way, check at the end: every side equal, every angle equal.

✦

Try it yourself

What angle do you measure at each vertex to draw a regular hexagon from its side?
What central angle do you use to draw a regular pentagon inside a circle?
Ishimwe drew a hexagon with all sides 4 cm, but his angles are 100°, 140°, 120°, 120°, 100° and 140°. Is it regular?
You draw a regular hexagon inside a circle of radius 3 cm. How long is each side?

Answers

120°
360° ÷ 5 = 72°
No. Its sides are equal, but its angles are not all equal.
3 cm, the same as the radius.

Quick check

You draw a regular pentagon from one side. What angle do you measure at each vertex?

What you learned today

Method 1: draw one side, then use the interior angle and equal sides to go round.
Method 2: draw a circle, mark central angles of 360° ÷ n, and join the points.
In a regular hexagon, each side equals the radius of the circle drawn round it.

Sources and credits

Mathematics Grade 8, 25.2 Regular polygons by Siyavula Education, licensed under CC BY 3.0. Adapted by Ganzaa.

The badge story, the circle-and-central-angle method (Method 2) and worked examples 2 and 3 were written by Ganzaa.

1Method 1: side and interior angle

2Method 2: the circle and the central angle

3Check your drawing

Try it yourself

Method 1: side and interior angle

Method 2: the circle and the central angle

Check your drawing