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# Circle problems

Area, volume, circles and Pythagoras · Circles

<!-- Speaker note: 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 (Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min). -->

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## Today

**We are learning to** 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.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> 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.

<!-- Speaker note: 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). -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A round rug has an area of 12.57 m². What is its radius?**

- 2 m
- 3.14 m
- 4 m
- 6.28 m

<!-- Speaker note: What happens: 2 m. 12.57 ÷ π = 4 = r², so r = √4 = 2 m. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask for the two steps of πr² in order. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fmeasurement%2Fcircle-problems%2Fstep-2&code=widget%3A+circlearea%0Aslider+r%3A+0.5%2C+6%2C+1%2C+0.5%0An%3A+16%0Aunits%3A+m%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/circle-problems/?full=step-2#step-2)

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask what area radius 1.5 gives, before moving the slider. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 2 m. 12.57 ÷ π = 4 = r², so r = √4 = 2 m.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Backwards

Here is the routine.

$$78.5 \div \pi \approx 25$$

1) $r = \sqrt{25} = 5\ \text{cm}$ — Undo the square with a square root.
2) $\pi \times 5^2 \approx 78.5$ — Check by going forwards.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 78.5 ÷ 25 and say "Undo the π: this is r²." Line 1: write r = 25 = 5cm and say "Undo the square with a square root." Line 2: write × 5^2 78.5 and say "Check by going forwards." -->

<!-- Speaker note: Teacher: ask Leo to check his radius forwards. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Backwards

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

* Answer: 6

<!-- Speaker note: 113.1 ÷ π ≈ 36, and √36 = 6 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

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

```
 1  A circle has an area of 50 cm².
 2  A = πr², so r = 50 ÷ 3.14.
 3  r = 15.9 cm.
```

* Answer: line 2

<!-- Speaker note: Line 2: 50 ÷ 3.14 is r², not r. Take the square root: r = √15.9 ≈ 4.0 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Semicircles

**How do you find the area of a semicircle of radius 4 cm?**

- Half of π × 4²
- π × 4² × 2
- Half of 2 × π × 4
- π × 2²

* Answer: Half of π × 4²

<!-- Speaker note: A semicircle is half the circle: ½ × π × 16 ≈ 25.1 cm². -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Tracks and rings



- **A semicircle:** 
- **A track's area:** 
- **A track's perimeter:** 
- **Over 100:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/circle-problems/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Composite problems with circles: a semicircle, a running track's area and perimeter, and a target area on the board. For students who need more support: A two-column card: 'length round the edge' (C = 2πr) and 'space inside' (A = πr²). For early finishers: Is a 30 cm pizza more than twice as much as a 20 cm pizza? Compare their areas and prices from a real menu. 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. -->

<!-- Speaker note: Teacher: ask how the two semicircles of the track make one circle. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Back to the radius

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

* Answer: 7

<!-- Speaker note: 154 ÷ π ≈ 49, and √49 = 7 m. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A quarter circle

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

* Answer: 78.5

<!-- Speaker note: ¼ × π × 100 ≈ 78.5 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A ring

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

* Answer: 50.3

<!-- Speaker note: π × 25 − π × 9 = π × 16 ≈ 50.3 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A semicircle's perimeter

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

* Answer: 25.7

<!-- Speaker note: Half the circumference, π × 5 ≈ 15.7 cm, plus the diameter 10 cm: 25.7 cm. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a garden bed

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

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/circle-problems/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing a radius to meet an area range uses the formula backwards as a prediction, checked on the board. Hints, in order: 25 ÷ π ≈ 8 and 30 ÷ π ≈ 9.5. / Which radius has a square between those? -->

<!-- Speaker note: Teacher: ask which formula the edging needs. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A circular pond has an area of 28.3 m². What is its radius?**

* 28.3 ÷ π ≈ 9, and √9 = 3 m.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This completes AC9M8M03. Where it leads: cylinders (Year 9), sectors and arc lengths (Year 10). -->