Because C ÷ d = π, the circumference is π × diameter: C = πd, or C = 2πr from the radius. Going backwards, d = C ÷ π.
A bike wheel 70 cm across rolls forward one circumference with every turn. How many turns to school? A trundle wheel measures a sports field the same way.
The last lesson found that C÷d=π for every circle. Turned round, that is a formula for the circumference:
C=πdor, since d=2r,C=2πr
Use π≈3.14, as the questions say. Going backwards, from a circumference to the diameter, divide by π: d=C÷π.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Radius 5
Find the diameter first.
TryTry it on the board and compare the result with your prediction.
Roll radius 2.5
Predict the circumference from the diameter before you roll.
Try it yourselfYour edits are saved in this browser.
Make a circle with radius 2.5 cm and roll it out. Work out π × d first, then compare. Was your prediction method right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Forwards and backwards
Here is the formula forwards and backwards.
Worked exampleBefore each step, say what you think comes next.
C=πd=3.14×8=25.12 cmFrom the diameter: multiply by π.
C=2πr=2×3.14×30=188.4 cmFrom the radius: double it first (2r is the diameter).
d=C÷π=18.84÷3.14=6 mBackwards: divide by π.
C=10π cmExact: leave π in the answer (for d = 10). 10π means 10 × π, about 31.4.
Now try one: diameter 8 cm.
Read Mei's working. Which length does C = πd need?
Your working can be short notes. Then choose the reason for the key step.
A circle has diameter 8 cm. Using π ≈ 3.14, what is its circumference, in cm?
Mei finds how far a wheel rolls in one turn. Click the line where her working goes wrong.
A circular garden bed has circumference 18.84 m. Using π ≈ 3.14, find its diameter. Write your steps.
Your working
A model solution
C=πd, so 18.84=3.14×dthe circumference is π times the diameter
d=18.84÷3.14=6 m divide by π to undo multiplying by π
Check your own working
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
25 ÷ 3.14 is about 8: that is the diameter.
Another hint
Radius 4.
Try it yourselfYour edits are saved in this browser.
Make and roll a circle with circumference about 25 cm (between 24.5 and 25.5).
Use the board. This ticks when you get there.
Using π ≈ 22/7, find the circumference of a circle with radius 7 cm.
A circle has circumference 31.4 cm. Using π ≈ 3.14, what is its radius, in cm?
PractiseA few on your own, to check you have it.
Circumference
Diameter 12 cm. Circumference, in cm (π ≈ 3.14)?
Radius 3 cm. Circumference, in cm (π ≈ 3.14)?
A bike wheel has diameter 70 cm. How far does the bike go in 100 turns, in metres (π ≈ 3.14)?
Review: a triangle has base 10 cm and height 6 cm. Its area, in cm²?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A trundle wheel
Then work out how many clicks it makes around a 400 m running track.
Need a hint?
One turn should roll exactly 1 m = 100 cm.
Another hint
C = πd, so d = 100 ÷ 3.14.
A trundle wheel clicks once each turn. What diameter, in cm to one decimal place, makes it click once every metre? (π ≈ 3.14)
For teachers
Students use C = πd and C = 2πr to find circumferences, check against circles rolled out on the board, go backwards with d = C ÷ π, and design a trundle wheel (AC9M7M03, node 3). About 45 minutes.
I can find a circle's circumference from its diameter or radius.
I can find a diameter or radius from a circumference.
I can give an exact answer in terms of π.
Before class
Do the Run board yourself first. Students need π as C ÷ d (previous lesson), d = 2r, and dividing decimals (a calculator is fine).
The one idea: C = πd, because that is what π means. The common mistake is using the radius in place of the diameter (Mei's mistake, a spiral of Parts of a circle). Rearranging to d = C ÷ π uses the inverse-operations idea from Algebra.
Running the lesson
Predict: students who chose 15.7 used the radius as the diameter.
Run: ask students to predict before rolling, then compare.
Investigate: share two routes for the wheel: C = π × 60, or C = 2 × π × 30.
Modify: for the semicircle, ask students to trace the whole outline with a finger.
Make: build a cardboard trundle wheel if there is time.
The hard part (on purpose)
Going backwards from a circumference to a radius takes two steps (divide by π, then halve). Ask students to write which length they have and which they want before they press any calculator keys.
When students get stuck
A student multiplies π by the radius (Mei's mistake). Ask: Is the given length across the whole circle or half way? Fix: Double the radius to get the diameter, or use C = 2πr.
A student multiplies by π when going backwards. Ask: Should the diameter be bigger or smaller than the circumference? Fix: Smaller: divide by π.
A student forgets units or mixes cm and m. Ask: What unit is your answer in? What does the question want? Fix: Convert at the end: 100 cm = 1 m.
Stretch and support
Stretch: The Earth's equator is about 40 000 km. Estimate the Earth's diameter. If a rope round the equator were made 1 m longer, how high above the ground would it float?
Support: Keep C = πd and d = 2r on the board; use 3.14 on a calculator (the π key gives answers a little different from 3.14).
Where it leads
This completes AC9M7M03.
Where it leads: the area of a circle and composite shapes with circles (Year 8), arc length (Year 10), and Pythagoras (AC9M8M06, next in this pathway).
What each step is for
radius 5: Committing to a circumference for radius 5 shows whether they double to the diameter first, or use the radius as if it were the diameter.
roll radius 2.5: Rolling a circle of a given radius checks C = πd = 2πr against the track it leaves.
forwards and backwards: C = πd uses the diameter (Mei uses the radius instead); C = 2πr is the same rule. Going backwards divides by π; students own the working and pick the key reason.
one change at a time: One change at a time: building a circumference on the board, using 22/7, an exact answer, a radius from a circumference (two steps), then a semicircle's perimeter.
circumference: Forwards from d and r, a real-world distance and a review, mixed; using the radius as the diameter is the planted slip.
a trundle wheel: Designing a trundle wheel that clicks every metre turns C = πd round: the diameter is 100 ÷ π, a real design problem with a sensible rounding.
Now you can
Find a circle's circumference from its diameter or radius.