Circle problems

Choose the formula by what you need: C = 2πr for a length round the edge, A = πr² for the space inside. Work backwards from an area with r = √(A ÷ π). Half and quarter circles, rings and tracks are parts of circles added to or taken from other shapes.

A running track, a round garden bed, a pizza deal: is a 30 cm pizza more than twice a 20 cm one? Circle problems need the right formula, sometimes run backwards.

Ask first: is the answer a length round the edge (circumference, $C = 2\pi r$ ) or the space inside (area, $A = \pi r^2$ )?

To find the radius from an area, undo each step: divide by $\pi$ , then take the square root.

r = \sqrt{A \div \pi}

For part of a circle, take the fraction: a semicircle is half the area $\frac{1}{2}\pi r^2$ , a quarter circle a quarter. Its perimeter is the curved part plus the straight edges. Say which value of π you used (the π button, or 3.14).

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.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

A round rug

What times itself, times π, makes 12.57?

A round rug has an area of 12.57 m². What is its radius?2 m3.14 m4 m6.28 m

TryTry it on the board and compare the result with your prediction.

Find the radius

Try it yourselfYour edits are saved in this browser.

Find the radius of the 12.57 m² rug. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Backwards and in parts

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$78.5 \div \pi \approx 25$ Undo the π: this is r².
$r = \sqrt{25} = 5\ \text{cm}$ Undo the square with a square root.
$\pi \times 5^2 \approx 78.5$ Check by going forwards.

Read Leo's working. Is 50 ÷ 3.14 the radius, or the radius squared?

A circle has an area of 113.1 cm². Find its radius in cm.

Leo finds the radius of a circle with area 50 cm². Click the line where his working goes wrong.

How do you find the area of a semicircle of radius 4 cm?Half of π × 4²π × 4² × 2Half of 2 × π × 4π × 2²

ModifyChange the problem on purpose and see what happens.

Tracks and rings

Try it yourselfYour edits are saved in this browser.

Find the area of a semicircle of radius 4 cm, in cm², to 1 decimal place.

A running track's field is a 100 m by 60 m rectangle with a semicircle on each 60 m end. What is its area in m²? (Round to the nearest whole number.)

What is the perimeter of that field in metres, to the nearest whole number?

Find the smallest radius (in steps of 0.5 m) whose area is more than 100 m².

Use the board. This ticks when you get there.

PractiseA few on your own, to check you have it.

Circle problems

A circular pool cover has an area of 154 m². Find its radius in m.

Find the area of a quarter circle of radius 10 cm, to 1 decimal place.

A ring is between circles of radius 5 cm and 3 cm. Find its area in cm², to 1 decimal place.

Find the perimeter of a semicircle of radius 5 cm (the curve and the diameter), to 1 decimal place.

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A garden bed

Then work out how much edging the bed needs (its circumference), and say which value of π you used.

Try it yourselfYour edits are saved in this browser.

Need a hint?

25 ÷ π ≈ 8 and 30 ÷ π ≈ 9.5.

Another hint

Which radius has a square between those?

A round garden bed must have an area between 25 and 30 m². Choose its radius.

Use the board. This ticks when you get there.

For teachers

Students choose between circumference and area, find a radius from an area, and solve problems with semicircles, quarter circles, rings and tracks (AC9M8M03, node 2). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can choose the circumference or the area formula for a problem.
I can find the radius from an area.
I can solve problems with parts of circles.

Before class

Do the Run board yourself first. Students need the area of a circle (previous lesson) and square roots.

The one idea: ask whether the answer is a length (C = 2πr) or a space (A = πr²); to go back from an area, divide by π and take the square root. The common mistake is stopping at r² (Leo's mistake).

Answers accept π from the calculator or 3.14; ask students to say which they used.

Wider: composite shapes with circles (a stadium, a keyhole), and circles in First Nations Australian weaving designs, where the context is shared respectfully with local knowledge.

Running the lesson

Predict: ask for the two steps of πr² in order.
Run: ask what area radius 1.5 gives, before moving the slider.
Investigate: ask Leo to check his radius forwards.
Modify: ask how the two semicircles of the track make one circle.
Make: ask which formula the edging needs.

The hard part (on purpose)

Working backwards from an area. Ask students to write the two steps of πr² forwards before they undo them.

When students get stuck

A student stops at r² when working backwards. Ask: Is 15.9 the radius, or the radius squared? Fix: Take the square root.
A student finds a semicircle's perimeter as half the circumference only. Ask: Walk round the semicircle: what straight part is there? Fix: Add the diameter.
A student uses the diameter in πr². Ask: Radius or diameter? Fix: Halve the diameter first.

Stretch and support

Stretch: Is a 30 cm pizza more than twice as much as a 20 cm pizza? Compare their areas and prices from a real menu.

Support: A two-column card: 'length round the edge' (C = 2πr) and 'space inside' (A = πr²).

Where it leads

This completes AC9M8M03.

Where it leads: cylinders (Year 9), sectors and arc lengths (Year 10).

What each step is for

a round rug: Committing to a radius from an area shows whether they undo π and the square, or divide only.
find the radius: Searching for the radius that gives the area shows the two steps that undo A = πr².
backwards and in parts: Undo π then the square (Leo forgets the square root); a semicircle is half the area.
tracks and rings: Composite problems with circles: a semicircle, a running track's area and perimeter, and a target area on the board.
circle problems: Radius from area, parts of circles, a ring and a semicircle's perimeter, mixed; forgetting the square root is the planted slip.
a garden bed: Choosing a radius to meet an area range uses the formula backwards as a prediction, checked on the board.

Now you can

Choose the circumference or the area formula for a problem.
Find the radius from an area.
Solve problems with parts of circles.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson18. Time zonesA city with a bigger UTC offset is ahead; a smaller offset is behind. Find the difference between the two offsets, then add or take it away; the day changes when you cross midnight. A flight's arrival time is the departure time plus the flight, changed to the local time where it lands.

Source: content/maths/measurement/circle-problems/lesson.md