Cut a circle into thin slices and lay them top to tail: they make a shape close to a rectangle, πr long and r high. So the area of a circle is A = πr². It is between 2r² and 4r², about 3.14 times r².
How much pizza is in a pizza? Cut it into slices, lay them out top to tail, and a rectangle appears.
Cut a circle into equal slices (sectors) and lay them out top to tail. Half the slices point up and half point down, so the shape is about half the circumference long (πr) and one radius high (r). The more slices, the closer it is to a rectangle. So
A=πr×r=πr2
A quick check: a circle fits inside a square of side 2r (area 4r2) and around a square of area 2r2, so its area is between 2r2 and 4r2: about 3.14r2. Do not mix it up with the circumference, C=2πr, which is a length.
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 3
Imagine the circle inside a square 6 cm by 6 cm.
TryTry it on the board and compare the result with your prediction.
Slices into a rectangle
Try it yourselfYour edits are saved in this browser.
Rearrange the pieces, then use at least 16 slices. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Why it works
Read Jo's working. Is 2πr an area or a length?
Jo finds the area of a circle of radius 5 cm. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
Squares and sizes
Need a hint?
Set the radius to 4, then press Compare with squares.
Try it yourselfYour edits are saved in this browser.
Compare a circle of radius 4 cm with the squares. Between which two areas is it?
Use the board. This ticks when you get there.
Find the area of a circle of radius 10 cm, in cm², to 1 decimal place.
Rearrange 32 slices of a circle. How close is the shape to a rectangle?
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Circle areas
Find the area of a circle of radius 7 cm, in cm², to 1 decimal place.
A circle has diameter 10 m. Find its area in m², to 1 decimal place.
Review: find the circumference of a circle of radius 5 cm, in cm, to 1 decimal place.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
An area of 50
Then explain why the area is not 3.14 × 4 = 12.56.
Try it yourselfYour edits are saved in this browser.
Need a hint?
πr² = 50, so r² is about 16.
Another hint
The square root of 16 is 4.
Find the radius (in steps of 0.5 cm) whose area is closest to 50 cm².
Use the board. This ticks when you get there.
For teachers
Students cut a circle into slices and rearrange them into a near-rectangle πr by r, compare it with squares of r², and use A = πr² (AC9M8M03, node 1). About 35 minutes.
I can say why a circle's area is between 2r² and 4r².
I can find the area from the radius or the diameter.
Before class
Do the Run board yourself first. If you can, cut a paper circle into 16 slices and lay them out.
The one idea: the slices make a shape πr long and r high, so A = πr². The common mistake is using 2πr (the circumference) for the area (Jo's mistake).
The board's squares show the bounds 2r² < A < 4r², a quick check on any answer.
Wider: A = ½ × C × r (the rectangle is half the circumference by the radius).
Running the lesson
Predict: ask for the outside square's area first.
Run: ask what the bumps do as the slices get thinner.
Investigate: ask Jo which formula measures round the edge.
Modify: ask how many times bigger a radius-10 circle is than radius 5.
Make: ask why r² comes before multiplying by π.
The hard part (on purpose)
Keeping area and circumference apart. Ask students to say what each formula measures (a length or a space) before they use it.
When students get stuck
A student uses 2πr for area. Ask: Is your answer a length or an amount of space? Fix: Area is πr² (square units); 2πr is the circumference.
A student uses the diameter in πr². Ask: Is that the radius or the diameter? Fix: Halve the diameter first.
A student works out (πr)² or π × 2r. Ask: Which part is squared? Fix: Only r is squared: π × r × r.
Stretch and support
Stretch: Show that A = ½ × C × r using the rearranged slices. Then use it to find the area of a circle with circumference 31.4 cm.
Support: A card with both formulas side by side, labelled length and area.
Where it leads
This starts AC9M8M03.
Where it leads: circle problems (next lesson), cylinders (Year 9) and sectors.
What each step is for
radius 3: Committing to an area for radius 3 shows whether they reason between 2r² and 4r², or mix area with circumference.
slices into a rectangle: Rearranging more and more slices shows the shape approaching a rectangle πr by r, which gives A = πr².
why it works: The long side is half the circumference; area and circumference are different formulas (Jo uses 2πr); 2r² < A < 4r² is a check.
squares and sizes: Comparing with squares, using the formula, and doubling the radius show how area grows with r².
circle areas: Area from the radius or diameter and a formula choice, mixed; using 2πr for area is the planted slip; the circumference review keeps the other formula separate.
an area of 50: Searching for a radius to hit an area uses πr² as a prediction before the board checks it.