Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8 Unit 3, lesson 5 (powers of negative numbers): what is (−5)²?
Answer
25. Squaring multiplies a number by itself, and a negative times a negative is positive: (−5) × (−5) = 25. - From this unit, lesson 3 (circumference): a circle has a diameter of 20 cm. What is its circumference? Use 3.14.
Answer
62.8 cm. 3.14 × 20 = 62.8. - From Year 7 Unit 7, lesson 3 (triangles and parallelograms): a parallelogram has a base of 12 cm and a height of 5 cm. What is its area?
Answer
60 cm². The area is base times height: 12 × 5 = 60.
Words used in this lesson
- area
- the amount of flat space inside a shape. For example: How much pizza there is.
- radius
- a straight line from the centre of a circle to its edge. For example: Half of the width of a pizza.
- squared
- multiplied by itself. For example: 6 squared is 36.
- sector
- a slice of a circle, like a slice of pizza. For example: One of the eight slices.
- ring
- the region between two circles with the same centre. For example: The grass round a pond.
- semicircle
- half of a circle. For example: The top of an arched tunnel.
Is a 30 cm pizza one and a half times as much?
At a pizza place in Naples, a 20 cm pizza and a 30 cm pizza are on the menu. The 30 cm pizza is 1.5 times as wide. Predict: is it 1.5 times as much pizza, about twice as much, or something else? Use the first figure to see where the area comes from, then the second to test your prediction.
The circle has radius 4 cm and is cut into 8 sectors. Half of its circumference is 3.14 × 4 = 12.56 cm, and its radius is 4 cm, so its area is 12.56 × 4 = 50.24 cm².
Show me the steps
- A circle with radius 4 cm is cut into 8 sectors. Half of its circumference is 3.14 × 4 = 12.56 cm.
- Press Rearrange the sectors. They fit into a wavy strip, 12.56 cm long and 4 cm high. A rectangle that size has area 12.56 × 4 = 50.24 cm².
- Move to 32 sectors. The wavy edges flatten and the strip is nearly a rectangle, 12.56 cm by 4 cm: the area is 50.24 cm².
- Set the radius to 3 cm. Half of the circumference is 3.14 × 3 = 9.42 cm, the height is 3 cm, and the area is 9.42 × 3 = 28.26 cm².
The area of a circle is A = π × r × r = πr², where r is the radius.
- Why: the sectors make a strip as long as half the circumference, π × r, and as high as the radius r. With more and thinner sectors the strip is as close to a rectangle as you like, and a rectangle that size has area π × r × r.
- Radius, not diameter: halve the diameter first. A diameter of 20 cm is a radius of 10 cm.
- Squared: r × r grows faster than r. Use the second figure to find what doubling the radius does to the area.
Back to the pizzas: they are compared below, with the second figure to check.
Two circles, one scale
Now set the second circle to be a copy of the first, then 1.5 times the radius, then double. What happens to the area each time, and does it match your prediction for the pizza?
The first circle has area 3.14 × 3 × 3 = 28.26 cm². The second has area 3.14 × 4 × 4 = 50.24 cm². The second radius is 1.33 times the first, and the second area is 1.78 times the first.
Show me the steps
- Radius 3 cm and radius 4 cm. The areas are 3.14 × 9 = 28.26 cm² and 3.14 × 16 = 50.24 cm². The second radius is about 1.33 times the first, and the second area is about 1.78 times the first.
- Set the radii to 10 cm and 15 cm: a 20 cm pizza and a 30 cm pizza. The areas are 314 cm² and 706.5 cm². The radius is 1.5 times as big and the area is 2.25 times as big.
- Set the radii to 2 cm and 4 cm. The second radius is double the first, and the area is four times as big: 12.56 cm² and 50.24 cm².
A common mistake: putting the diameter into the area formula. A circle with a diameter of 4 cm has a radius of 2 cm, so its area is 3.14 × 2 × 2 = 12.56 cm². Using 4 gives 3.14 × 4 × 4 = 50.24 cm², four times too big. Always halve the diameter first. If the question says “diameter”, write the radius on the diagram before you square anything.
Worked examples
The pizzas: diameters of 20 cm and 30 cm. Find the area of each and say how many times as much the bigger one is. Use 3.14 for π.
- Radii: 10 cm and 15 cm.
- Areas: 3.14 × 10 × 10 = 314 cm², and 3.14 × 15 × 15 = 706.5 cm².
- 706.5 ÷ 314 = 2.25 times as much. That is 1.5 × 1.5, the ratio of the radii squared.
A round rug has a diameter of 3 m. Find its area. Use 3.14 for π.
- The radius is 3 ÷ 2 = 1.5 m.
- Area = 3.14 × 1.5 × 1.5 = 3.14 × 2.25.
- 3.14 × 2.25 = 7.065 m².
A circle has a radius of 9 cm. Give its area in terms of π, and as a decimal using 3.14.
- Area = π × 9 × 9.
- In terms of π: 81π cm².
- With 3.14: 3.14 × 81 = 254.34 cm².
Your go. The first step is given. A round pond has a diameter of 26 m. Find its area. Use 3.14 for π.
- The radius is half of the diameter: 26 ÷ 2 = 13 m.
- Now square the radius, multiply by 3.14, then check the size against three squares of side 13.
Check the full answer
13 × 13 = 169, and 3.14 × 169 = 530.66 m². Check: three squares of side 13 m are 507 m², and 530.66 is a little more.
With a calculator
With the π key, the area of a circle with radius 20 cm is π × 20² = 1256.637…, which rounds to 1256.6 cm². With 3.14 it is 3.14 × 20 × 20 = 1256 cm². The questions here use 3.14 so that every answer is exact. Square only the radius: π × 20² is not (π × 20)².
Estimate first: a circle's area is a little over three squares on its radius. For radius 20 cm, three squares of 400 cm² make 1200 cm², so 1256 cm² is about right. An answer near 63 or near 5000 is a slip with the radius or the diameter.
Check before you go on
1. Which calculation gives the area of this circle 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 circular flower bed has a radius of 6 m. What is its area in square metres? Use 3.14 for π.
3. A circle has a radius of 7 cm. Its area is __ π cm². What number goes in the gap?
Level 2 · Building
4. A circular pond in Accra has a diameter of 14 m. What is its area in square metres? Use 3.14 for π.
5. A circle has an area of 78.5 cm². What is its radius in centimetres? Use 3.14 for π.
6. A circular lawn in Kuala Lumpur has a round pond in the middle. How many square metres of grass are there? Use 3.14 for π.
Level 3 · Stretch
7. The cross-section of a tunnel is a rectangle 6 m wide and 3 m tall with a semicircle on top, as in the diagram. What is the area of the cross-section in square metres? Use 3.14 for π.
8. A circular tray has a radius of 8 cm. A bigger tray, the same shape, has a radius of 20 cm. How many times as big is the area of the bigger tray?
Something to think about
For most circles the number of the circumference in centimetres and the number of the area in square centimetres are different. Is there a circle where they are the same number? (The units are different, but ask whether the numbers match.)
One way to think about it
Yes, radius 2 cm. The circumference is 2 × 3.14 × 2 = 12.56 and the area is 3.14 × 2 × 2 = 12.56. Setting 2 × π × r equal to π × r × r and dividing both sides by π × r gives 2 = r. It is a coincidence of the numbers: a length and an area cannot really be equal, and a different unit of length would break it.
Exit ticket
9. A circle has a radius of 30 cm. Use 3.14 for π. Which pair gives its circumference and its area?