Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7 Unit 4, lesson 2 (angles at a point): five equal angles meet at a point. How big is each one?
Answer
72°. All the angles at a point add to 360°, and 360 ÷ 5 = 72. - From Year 7 Unit 7, lesson 3 (triangles and parallelograms): a triangle has a base of 8 cm and a height of 5 cm at a right angle to it. What is its area?
Answer
20 cm². Half of 8 × 5 = 40 is 20. - From this unit, lesson 1 (area of a trapezium): a trapezium has parallel sides of 5 cm and 7 cm and a height of 4 cm. What is its area?
Answer
24 cm². ½ × (5 + 7) × 4 = 6 × 4 = 24.
Words used in this lesson
- regular polygon
- a flat shape with equal sides and equal angles. For example: A square, or the face of a stop sign.
- apothem
- the distance from the centre to the middle of a side, at a right angle to it. For example: The height of one of the triangles.
- perimeter
- the distance all the way round a shape. For example: The fence round a field.
- centre
- the point in the middle of a regular polygon. For example: Where the triangles meet.
- hexagon
- a polygon with six sides. For example: A honeycomb cell.
- identical
- exactly the same shape and size. For example: The six triangles in a hexagon.
An octagon is not a rectangle
Hind cuts regular octagonal tiles in a workshop in Marrakech. Each side is 5 cm, all eight sides are the same, and all the angles are the same. She needs the area of one face to order glaze. There is no length-times-width here. Predict: if you join the centre of the tile to each of its corners, how many triangles do you get, and what can you say about them? Then use the sliders and press Show the triangles.
The drawing is resized to fill the frame, so compare the numbers, not the sizes.
A regular hexagon with side 4 cm. Its apothem is about 3.5 cm, so each triangle has area ½ × 4 × 3.5 = 7 cm² and the hexagon is about 6 × 7 = 42 cm².
Show me the steps
- A regular hexagon has side 4 cm. Its apothem is about 3.5 cm. The perimeter is 6 × 4 = 24 cm and the area is about 6 × ½ × 4 × 3.5 = 42 cm².
- Press Show the triangles. Six identical triangles meet at the centre, each with an angle of 360 ÷ 6 = 60° there, base 4 cm and height 3.5 cm. Each has area ½ × 4 × 3.5 = 7 cm², and 6 × 7 = 42 cm².
- Make it 8 sides with side 4 cm. The apothem is about 4.8 cm, each triangle has area ½ × 4 × 4.8 = 9.6 cm², and the area is about 8 × 9.6 = 76.8 cm².
- Make it 4 sides, a square with side 4 cm. The apothem is 2 cm, each triangle has area 4 cm², and 4 × 4 = 16 cm², the same as the square's 4 × 4.
- Make it 3 sides with side 5 cm. The apothem is about 1.4 cm and the area is about 3 × ½ × 5 × 1.4 = 10.5 cm². The true area is slightly more, because 1.4 is rounded down.
A regular polygon with n sides of length s and apothem a has area n × ½ × s × a. The apothem is the distance from the centre to the middle of a side, measured at a right angle to the side.
- Why: joining the centre to the corners gives n identical triangles, each with base s and height a. One triangle is ½ × s × a.
- In terms of the perimeter: the perimeter is n × s, so the area is also ½ × perimeter × a.
- The angle at the centre of each triangle is 360° ÷ n, because the angles at a point add to 360°.
Back to the tile: n = 8, s = 5, and the apothem is worked out for you below.
Lay the triangles in a row
The same triangles can be turned alternately and laid in a row. What shape do you get when the number of sides is even, and what when it is odd? What does its area have to be?
A regular hexagon with side 3 cm is 6 triangles, each ½ × 3 × 2.6 = 3.9 cm². Together: 6 × 3.9 = 23.4 cm².
Show me the steps
- A regular hexagon with side 3 cm is six triangles, each ½ × 3 × 2.6 = 3.9 cm². Together they have area 6 × 3.9 = 23.4 cm².
- Press Lay the triangles in a row. With 6 triangles the row is a parallelogram with base 3 × 3 = 9 cm, half of the perimeter 18 cm, and height 2.6 cm. Its area is 9 × 2.6 = 23.4 cm².
- Make it 5 sides. The row is a trapezium with parallel sides 3 × 3 = 9 cm and 2 × 3 = 6 cm. Their average is 7.5 cm, which is half of the perimeter 15 cm.
- With 5 sides and side 3 cm the apothem is about 2.1 cm, so the area is 7.5 × 2.1 = 15.75 cm², the same as 5 × ½ × 3 × 2.1.
A common mistake: using the distance from the centre to a corner as the height. It is longer than the apothem and it is not at a right angle to the side. In a regular hexagon with side 4 cm the corner is 4 cm from the centre, but the apothem is about 3.5 cm. Using 4 gives 6 × ½ × 4 × 4 = 48 cm², which is too big. With the apothem it is 6 × ½ × 4 × 3.5 = 42 cm². Look for the small square at the middle of the side.
Worked examples
A regular hexagon has sides of 10 cm and an apothem of 8.7 cm. Find its area.
- Six identical triangles, each with base 10 cm and height 8.7 cm.
- One triangle: ½ × 10 × 8.7 = 43.5 cm².
- Six of them: 6 × 43.5 = 261 cm².
A regular nonagon has a perimeter of 54 cm and an apothem of 8.2 cm. Find its area.
- The perimeter is the sides added up, so the area is ½ × perimeter × apothem.
- ½ × 54 = 27, and 27 × 8.2 = 221.4 cm².
- You never needed the number of sides or the length of one side.
Hind's tile: a regular octagon with sides of 5 cm and an apothem of about 6 cm. Find the area of one face.
- Eight triangles, each with base 5 cm and height 6 cm.
- One triangle: ½ × 5 × 6 = 15 cm².
- Eight of them: 8 × 15 = 120 cm², about 120 cm² because the apothem is rounded.
Your go. The first step is given. A regular pentagon has sides of 10 cm and an apothem of 6.9 cm. Find its area.
- Five identical triangles, each with base 10 cm and height 6.9 cm.
- Now find the area of one triangle, then multiply by 5, then check.
Check the full answer
One triangle: ½ × 10 × 6.9 = 34.5 cm². Five: 5 × 34.5 = 172.5 cm². Check with the perimeter: ½ × 50 × 6.9 = 25 × 6.9 = 172.5.
With a calculator
Enter a whole area in one go, with the brackets the structure suggests. For a pentagon with side 12 cm and apothem 8.3 cm, the keys are 5 × 0.5 × 12 × 8.3, which gives 249. Estimate first: about 5 × 6 × 8 = 240, so 249 is believable.
The apothems on this page are rounded to one decimal place, because the true value is a long decimal. The areas you find are therefore close to, not exactly equal to, the true areas. Use the numbers given in each question.
Check before you go on
1. Which calculation gives the area of this regular pentagon in square centimetres?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A regular polygon has 9 sides. It is cut into triangles by lines from its centre to the corners. How big is the angle at the centre of each triangle, in degrees?
3. One of the identical triangles in a regular pentagon has a base of 7 cm and a height of 4.8 cm. What is its area in square centimetres?
Level 2 · Building
4. A regular hexagon has sides of 6 cm and an apothem of 5.2 cm. What is its area in square centimetres?
5. Find the area of this regular decagon in square centimetres.
6. A regular octagon has sides of 10 cm and an area of 484 cm². What is its apothem in centimetres?
Level 3 · Stretch
7. A gazebo in Colombo has a floor that is a regular hexagon with sides of 2 m and an apothem of 1.7 m. Tiling costs 500 rupees for each square metre. How many rupees does it cost to tile the floor?
8. A regular polygon has a perimeter of 72 cm and an apothem of 10.4 cm. What is its area in square centimetres?
Something to think about
A fence is 60 m long and goes all the way round a regular polygon, so the perimeter is 60 m. Compare a hexagon with sides of 10 m (apothem about 8.7 m), a decagon with sides of 6 m (apothem about 9.2 m) and a 12-sided polygon with sides of 5 m (apothem about 9.3 m). Which encloses the most ground, and what do you think happens as the number of sides keeps growing?
One way to think about it
Hexagon: ½ × 60 × 8.7 = 261 m². Decagon: ½ × 60 × 9.2 = 276 m². 12 sides: ½ × 60 × 9.3 = 279 m². More sides, more ground, with the same fence. The gains get smaller each time, and the shape gets closer to a circle. A circle with a 60 m circumference holds about 286 m², which you will find in lesson 4.
Exit ticket
9. A regular hexagon has sides of 8 cm. Each corner is 8 cm from the centre, and the apothem is about 6.9 cm. Which calculation gives its area?