Regular Polygons and Bearings
Primary 6 · Mathematics
Regular Polygons and Bearings
Lesson 2 of 5
18 min
Let's begin
Ishimwe's school is building a flower bed shaped like a regular pentagon in front of the office. The builder has cut five equal pieces of timber for the sides, but he has a question: at what angle should I join each corner? If the angles are wrong, the last side will not meet the first one.
Today you will learn to work out the angles of any regular polygon using just two numbers you already know: 180° and 360°.
Choose one vertex and draw straight lines from it to every other vertex that is not next to it. The polygon is cut into triangles, and the angles of each triangle add up to 180°.
| Polygon | Sides (n) | Triangles (n − 2) | Sum of interior angles |
|---|---|---|---|
| triangle | 3 | 1 | 180° |
| quadrilateral | 4 | 2 | 360° |
| pentagon | 5 | 3 | 540° |
| hexagon | 6 | 4 | 720° |
| heptagon | 7 | 5 | 900° |
| octagon | 8 | 6 | 1 080° |
| nonagon | 9 | 7 | 1 260° |
| decagon | 10 | 8 | 1 440° |
A polygon with n sides always makes (n − 2) triangles, so:
sum of interior angles = (n − 2) × 180°
In a regular polygon all the interior angles are equal, so share the sum out equally:
each interior angle = (n − 2) × 180° ÷ n
Worked example 1
the pentagon flower bed.
Worked example 2
a regular octagon.
At every vertex, the interior angle and the exterior angle sit side by side on a straight line, so together they make 180°. Two angles that add up to 180° are called supplementary. (Two angles that add up to 90° are called complementary: for example, 60° and 30°.)
Now imagine walking all the way round the edge of a polygon. At each corner you turn through the exterior angle. When you are back at the start, facing the same way, you have turned exactly once round: 360°. So:
Worked example 3
a regular hexagon.
Join the centre of a regular polygon to every vertex. The n angles at the centre are all equal, and together they make a full turn, so each central angle = 360° ÷ n. In a regular polygon this is the same size as the exterior angle. The regular pentagon below has central angles of 360° ÷ 5 = 72° and interior angles of 108°.
Worked example 4
each interior angle of a regular polygon is 140°. How many sides does it have?
If 360 ÷ (exterior angle) is not a whole number, then no regular polygon has that angle.
Tip
Start from the outside. The outside turns always make one full turn, 360°. So work out exterior = 360 ÷ n first, then interior = 180 − exterior. Two small sums, and you never get lost.
Answers
What you learned today
Sources and credits
The triangle method for angle sums, the "walk around the polygon" explanation of exterior angles and all the worked examples were written by Ganzaa.
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