Central Angles: From Heptagons to Decagons

18 min

Let's begin

Look at the clock on your classroom wall. The 12 hour marks sit evenly round a circle, and the angle at the centre between one mark and the next is 360° ÷ 12 = 30°.

That angle has a name: it is a central angle. Once you can work it out and mark it, you can construct any regular polygon, even the tricky seven-sided heptagon.

The central angle

For a regular polygon with n sides: central angle = 360° ÷ n.

Polygon	Pentagon	Hexagon	Heptagon	Octagon	Nonagon	Decagon
Sides (n)	5	6	7	8	9	10
Central angle	72°	60°	about 51.4°	45°	40°	36°

The central angle and the interior angle always add up to 180°. For an octagon: 45° + 135° = 180°.

Worked example 1: a regular octagon, with compasses and a ruler only

Draw a circle with centre O and draw a diameter AE.
Construct a second diameter CG at right angles to AE (use the right-angle construction from the last lesson at O). You now have 4 points, 90° apart.
Bisect (cut in half) each right angle: with the point on O, draw an arc crossing both arms. From those two crossing points, draw equal arcs that cross each other. The line from O through that crossing splits 90° into two angles of 45°.
Extend the four new lines to meet the circle. You now have 8 points, 45° apart.
Join the 8 points in order to make a regular octagon.

Worked example 2: a regular decagon in a circle of radius 5 cm

Central angle = 360° ÷ 10 = 36°.
Draw the circle and one radius OA. Measure 36° from OA with a protractor and draw a second radius, meeting the circle at B.
Open your compasses to the distance AB. Step this distance round the circle from B, marking each point.
After 10 steps you should be back at A. Join the points in order.

Heptagons and nonagons

For a nonagon, the central angle is 360° ÷ 9 = 40°. For a heptagon, 360 ÷ 7 does not divide exactly: it is about 51.4°. A protractor cannot mark 51.4° perfectly, so measure it once as carefully as you can, then copy the side round the circle with your compasses. If the last step misses the start by a little, adjust the compasses very slightly and try again.

Worked example 3: working backwards

Byiringiro marks central angles of 45° all the way round a circle. How many points does he get? 360° ÷ 45° = 8, so joining them makes a regular octagon.

Tip

Central angle = 360 ÷ number of sides. Measure it once, then let the compasses copy the side all the way round. The compasses are more accurate than the protractor.

✦

Try it yourself

What is the central angle of a regular decagon?
A regular polygon has a central angle of 40°. How many sides does it have?
The central angle of a regular octagon is 45°. What is its interior angle?
Why is a regular heptagon harder to construct than a regular hexagon?

Answers

360° ÷ 10 = 36°.
360 ÷ 40 = 9 sides (a nonagon).
180° − 45° = 135°.
Its central angle, 360° ÷ 7, is about 51.4°, which is not a whole number of degrees. A hexagon's is exactly 60°, and its side equals the radius.

Quick check

Mutesi wants to draw a regular polygon with 12 sides inside a circle. What central angle does she mark?

What you learned today

The central angle of a regular polygon is 360° ÷ n, and central angle + interior angle = 180°.
You can construct an octagon by bisecting right angles, and a decagon from central angles of 36°.
Measure the central angle once, then step the side round the circle with your compasses.

Sources and credits

Mathematics Grade 8, 25.2 Regular polygons by Siyavula Education, licensed under CC BY 3.0. Adapted by Ganzaa.
Mathematics Grade 7, Chapter 4: Construction of geometric figures by Siyavula Education, licensed under CC BY 3.0. Adapted by Ganzaa.

The central-angle method, the octagon and decagon construction steps and the worked examples were written by Ganzaa.

1The central angle