Regular Polygons and Bearings
Primary 6 · Mathematics
Regular Polygons and Bearings
Lesson 3 of 5
18 min
Let's begin
Umutoni's aunt runs a small café in Musanze. She wants new tables with hexagon-shaped tops, and hexagon tiles on the floor. Two questions come up at once: how much edging strip goes round one table top? and will hexagon tiles cover the floor with no gaps?
This lesson answers both questions, and shows you how to find the area of a regular polygon too.
The perimeter is the distance all the way round. A regular polygon has n equal sides, so:
perimeter = n × length of one side
Worked example 1
a regular hexagonal table top has sides of 50 cm. Perimeter = 6 × 50 cm = 300 cm, which is 3 m of edging strip.
Worked example 2
(backwards): a regular pentagon has a perimeter of 45 m. One side = 45 m ÷ 5 = 9 m.
Worked example 3
(how many sides?): a regular polygon has a perimeter of 72 cm, and each side is 9 cm. Number of sides = 72 ÷ 9 = 8, so it is a regular octagon.
The apothem is the straight distance from the centre of a regular polygon to the middle of one side. It meets that side at a right angle. Think of it as the height of one of the triangles you get when you join the centre to every vertex.
You can find the area of many shapes by cutting them into triangles. Join the centre of a regular polygon to every vertex, and it splits into n identical triangles (look at the hexagon below).
Worked example 4
a regular hexagon has sides of 6 cm and an apothem of 5.2 cm.
Worked example 5
(test the rule on a square): a square of side 12 cm has an apothem of 6 cm, which is half of a side.
Worked example 6
(finding the apothem): a regular hexagon has a perimeter of 60 cm and an area of 259.8 cm². How long is its apothem?
Tiles fit together with no gaps only if the angles that meet at each corner make exactly 360°. For tiles that are all the same regular polygon, the interior angle must divide 360° exactly.
| Tile | Interior angle | 360° ÷ angle | Tiles a floor on its own? |
|---|---|---|---|
| equilateral triangle | 60° | 6 | Yes: 6 meet at each point |
| square | 90° | 4 | Yes: 4 meet at each point |
| regular pentagon | 108° | about 3.3 | No: gaps are left |
| regular hexagon | 120° | 3 | Yes: 3 meet at each point |
| regular octagon | 135° | about 2.7 | No: gaps are left |
Only three regular polygons tile a flat floor on their own: the equilateral triangle, the square and the regular hexagon. Bees build their honeycomb from hexagons, and Umutoni's aunt can use hexagon tiles too.
Tip
Cut the cake. A regular polygon is like a cake cut from the middle into equal slices, and every slice is a triangle. Area = ½ × perimeter × apothem simply adds all the slices together.
For tiling, ask one question: does the corner angle go into 360 exactly?
Answers
What you learned today
Sources and credits
The café story, the apothem formula, the tiling table and all the worked examples were written by Ganzaa.
Quick check
A regular hexagon has sides of 8 cm. What is its perimeter?
