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 7, lesson 1 (perimeter and area of rectangles): a rectangle is 12 cm by 7 cm. What is its perimeter?
Answer
38 cm. 12 + 7 + 12 + 7 = 38, or 2 × (12 + 7). - From Year 7 Unit 1, lesson 7 (rounding and estimating): round 3.14159 to two decimal places.
Answer
3.14. The third decimal place is 1, so the 4 stays. - From Year 8 Unit 4, lesson 3 (solving equations by balancing): solve 5x = 90.
Answer
x = 18. Divide both sides by 5: 90 ÷ 5 = 18.
Words used in this lesson
- circumference
- the distance all the way round a circle. For example: The length of string round a tin.
- diameter
- a straight line across a circle through the centre. For example: The width of a wheel.
- radius
- a straight line from the centre to the edge, half of the diameter. For example: A spoke of a wheel.
- π
- the number you get when you divide a circle's circumference by its diameter, about 3.14. For example: 3.14 is a rounded value of π.
- in terms of π
- an answer written with the symbol π in it, not as a decimal. For example: Radius 3 cm gives a circumference of 6π cm.
- semicircle
- half of a circle. For example: The top of an arched window.
How far does the bicycle go in one turn?
Linh rides to school in Hanoi on a bicycle with wheels 50 cm across. Predict: how far does the bicycle go forward in one full turn of a wheel: about 100 cm, about 150 cm, or about 200 cm? Then use the slider and press Roll the edge out, and see what happens to the circle's edge.
The circle has diameter 6 cm, so its circumference is 3.14 × 6 = 18.84 cm. The circumference divided by the diameter is 18.84 ÷ 6 = 3.14.
Show me the steps
- A circle with diameter 6 cm has circumference 3.14 × 6 = 18.84 cm. Divide by the diameter: 18.84 ÷ 6 = 3.14.
- Press Roll the edge out. The edge laid flat is 3 whole diameters and a short piece, 0.14 of a diameter, left over.
- Set the diameter to 10 cm: 3.14 × 10 = 31.4 cm, and 31.4 ÷ 10 = 3.14 again.
- Set the diameter to 3 cm: 3.14 × 3 = 9.42 cm, and 9.42 ÷ 3 = 3.14. Every size gives the same number, which is about 3.14.
The circumference of a circle is C = π × d = 2 × π × r, where d is the diameter, r is the radius and π is about 3.14.
- Where π comes from: every circle is the same shape, only bigger, so the edge is the same number of diameters long. That number is π: just over 3, and its decimals never end or repeat. 3.14 is a rounded value.
- Diameter and radius: d = 2 × r. A formula with d takes the whole width; a formula with r takes half of it.
- In terms of π: leave π as a symbol and the answer is exact: radius 5 cm gives C = 10π cm.
Back to Linh: the wheels are 50 cm across, and the distance for one turn is worked out below.
A wheel rolls its circumference
When a wheel makes one full turn, each point on its edge touches the ground once, so it rolls forward exactly its circumference. Roll it for several turns and then change its size.
The wheel has diameter 40 cm, so one turn is 3.14 × 40 = 125.6 cm. In 2 turns it rolls 2 × 125.6 = 251.2 cm.
Show me the steps
- The wheel has diameter 40 cm. One turn is 3.14 × 40 = 125.6 cm, so two turns cover 2 × 125.6 = 251.2 cm.
- Set 5 turns. The distance is 5 × 125.6 = 628 cm, which is 6.28 m.
- Set the diameter to 70 cm and 4 turns. One turn is 3.14 × 70 = 219.8 cm, and 4 turns cover 4 × 219.8 = 879.2 cm.
A common mistake: using the radius where the formula wants the diameter. For a circle with radius 5 cm, 3.14 × 5 = 15.7 cm is only half of the edge. The edge must be a little more than three diameters, and the diameter is 10 cm, so it must be a little over 30 cm. The right calculation is 3.14 × 10 = 31.4 cm, or 2 × 3.14 × 5 = 31.4 cm.
Worked examples
Linh's bicycle has wheels with a diameter of 50 cm. How far does it go in one turn of a wheel? Use 3.14 for π.
- The diameter is 50 cm. The distance for one turn is the circumference.
- C = 3.14 × 50 = 157 cm.
- Check: three diameters are 150 cm, and a little more is 157 cm.
A circle has a radius of 7 cm. Find its circumference, and write it in terms of π.
- The diameter is 2 × 7 = 14 cm.
- In terms of π: C = 14π cm.
- With 3.14: C = 3.14 × 14 = 43.96 cm.
The circumference of a circle is 62.8 cm. Find its diameter and its radius.
- C = 3.14 × d, so 3.14 × d = 62.8.
- Divide both sides by 3.14: d = 62.8 ÷ 3.14 = 20 cm.
- The radius is half of that: 10 cm.
Your go. The first step is given. A circular pond has a diameter of 30 m. How far is it round the edge? Use 3.14 for π.
- The diameter is 30 m, so use C = 3.14 × d.
- Now multiply, then check the size against three diameters.
Check the full answer
C = 3.14 × 30 = 94.2 m. Check: three diameters are 90 m, and 94.2 is a little more.
With a calculator
Your calculator has a π key. It holds 3.14159265… and keeps going. This lesson uses 3.14 in every question, so that everyone gets the same answer and the answers can be checked exactly. For a circle with diameter 10 cm, 3.14 gives 31.4 cm and the π key gives 31.42 cm (to two decimal places). Both are right to the accuracy of the measuring.
Estimate before you press anything: a circle's edge is a little over three diameters, so a diameter of 10 cm should give a little over 30 cm. An answer near 15 or near 100 means a slip.
Check before you go on
1. Which calculation gives the circumference of this circle in 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 round table in Cairo has a diameter of 90 cm. A strip of edging goes all the way round the edge. How long is the strip, in centimetres? Use 3.14 for π.
3. A circle has a radius of 8 cm. Its circumference is __ π cm. What number goes in the gap?
Level 2 · Building
4. A round tin lid has a circumference of 25.12 cm. What is its diameter in centimetres? Use 3.14 for π.
5. A bicycle wheel has a diameter of 60 cm. How many complete turns does the wheel make while the bicycle travels 56.52 m? Use 3.14 for π.
6. The frame of this window has a rectangle 80 cm wide and 120 cm tall with a semicircle on top. The frame goes up both tall sides, along the bottom and round the curve. How long is the frame, in centimetres? Use 3.14 for π.
Level 3 · Stretch
7. A circular pond in Accra has a radius of 20 m. A path 2 m wide goes all the way round it. How many metres longer is the outer edge of the path than the edge of the pond? Use 3.14 for π.
8. Circle A has a diameter of 7 cm. Circle B has a diameter of 14 cm. How many centimetres longer is the circumference of circle B than the circumference of circle A? Use 3.14 for π.
Something to think about
A string is tied tight round a football, and another is tied tight round the Earth's equator. You add exactly 1 m to each string and lift it so that it floats the same height above the surface all the way round. Which gap is bigger: the one round the football or the one round the Earth?
One way to think about it
They are the same, about 16 cm. Adding 1 m (100 cm) to the circumference adds 100 ÷ (2 × 3.14) = 15.9 cm to the radius, and the size of the circle never appears. The Earth is enormous and the football is small, but the extra length is shared round the circle in the same way. You can check with the second figure's idea: the circumference grows 3.14 cm for every 1 cm of diameter.
Exit ticket
9. Circle P has a diameter of 16 cm. Circle Q has a diameter of 48 cm. Use 3.14 for π. Which sentence is right?