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# Area of a circle

Area, volume, circles and Pythagoras · Circles

<!-- Speaker note: 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 (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** 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.

- I can explain why the area of a circle is πr².
- I can say why a circle's area is between 2r² and 4r².
- I can find the area from the radius or the diameter.

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

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## Why it matters

> How much pizza is in a pizza? Cut it into slices, lay them out top to tail, and a rectangle appears.

<!-- Speaker note: 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 ($\pi r$) and **one radius** high ($r$). The more slices, the closer it is to a rectangle. So $$A = \pi r \times r = \pi r^2$$ A quick check: a circle fits inside a square of side $2r$ (area $4r^2$) and around a square of area $2r^2$, so its area is between $2r^2$ and $4r^2$: about $3.14 r^2$. Do not mix it up with the circumference, $C = 2\pi r$, which is a length. -->

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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 circle has radius 3 cm. Which is closest to its area?**

- 28 cm²
- 19 cm²
- 36 cm²
- 9 cm²

<!-- Speaker note: What happens: 28 cm². It is between 2 × 9 = 18 and 4 × 9 = 36 cm²: about 3.14 × 9 = 28.3 cm². (19 cm² is close to the circumference, 18.8 cm, a length.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask for the outside square's area first. -->

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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-area%2Fstep-2&code=widget%3A+circlearea%0Aslider+r%3A+1%2C+5%2C+3%0Aslider+n%3A+4%2C+32%2C+8%2C+4%0Acut%3A+on%0Aunits%3A+cm%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask what the bumps do as the slices get thinner. -->

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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?

* 28 cm². It is between 2 × 9 = 18 and 4 × 9 = 36 cm²: about 3.14 × 9 = 28.3 cm². (19 cm² is close to the circumference, 18.8 cm, a length.)

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

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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>

## The long side

**Why is the rearranged shape πr long?**

- Because there are n slices
- Because π is about 3
- Half the slices' curved edges are along the bottom: half of the circumference 2πr
- Because the circle has radius r

* Answer: Half the slices' curved edges are along the bottom: half of the circumference 2πr

<!-- Speaker note: The curved edges of the slices make the circumference; half is on top, half on the bottom: πr each. -->

<!-- Speaker note: Teacher: ask Jo which formula measures round the edge. -->

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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

**Jo finds the area of a circle of radius 5 cm. Click the line where her working goes wrong.**

```
 1  A circle has radius 5 cm.
 2  Its area is 2 × π × 5.
 3  2 × 3.14 × 5 = 31.4 cm².
```

* Answer: line 2

<!-- Speaker note: Line 2: 2πr is the circumference (a length). The area is πr² = 3.14 × 25 ≈ 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 check with squares

**A circle has radius 5 cm. Why must its area be between 50 and 100 cm²?**

- It is more than the inside square (2r² = 50) and less than the outside square (4r² = 100)
- Because 5 × 10 = 50
- Because π is between 2 and 4 exactly
- Because the diameter is 10

* Answer: It is more than the inside square (2r² = 50) and less than the outside square (4r² = 100)

<!-- Speaker note: The outside square has side 2r (area 4r² = 100); the square on the diagonals has area 2r² = 50. -->

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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>

## Squares and sizes



- **Squares:** 
- **Radius 10:** 
- **Double the radius:** 
- **32 slices:** 

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

<!-- Speaker note: What this step is for: Comparing with squares, using the formula, and doubling the radius show how area grows with r². Hints, in order: Set the radius to 4, then press Compare with squares. For students who need more support: A card with both formulas side by side, labelled length and area. For early finishers: 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. 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. -->

<!-- Speaker note: Teacher: ask how many times bigger a radius-10 circle is than radius 5. -->

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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>

## Radius 7

**Find the area of a circle of radius 7 cm, in cm², to 1 decimal place.**

* Answer: 153.9

<!-- Speaker note: π × 49 ≈ 153.9 cm² (with 3.14: 153.86). -->

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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>

## From the diameter

**A circle has diameter 10 m. Find its area in m², to 1 decimal place.**

* Answer: 78.5

<!-- Speaker note: r = 5 m, so A = π × 25 ≈ 78.5 m². -->

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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>

## The formula

**Which formula gives the area of a circle?**

- A = πr²
- A = 2r²
- A = πd
- A = 2πr

* Answer: A = πr²

<!-- Speaker note: A = πr²; the circumference is C = 2πr = πd. -->

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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>

## Review: circumference

**Review: find the circumference of a circle of radius 5 cm, in cm, to 1 decimal place.**

* Answer: 31.4

<!-- Speaker note: C = 2 × π × 5 ≈ 31.4 cm. -->

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## Make: an area of 50

Then explain why the area is not 3.14 × 4 = 12.56.

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

<!-- Speaker note: What this step is for: Searching for a radius to hit an area uses πr² as a prediction before the board checks it. Hints, in order: πr² = 50, so r² is about 16. / The square root of 16 is 4. -->

<!-- Speaker note: Teacher: ask why r² comes before multiplying by π. -->

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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

**Find the area of a circle with diameter 8 cm.**

* r = 4 cm; A = π × 16 ≈ 50.3 cm².

<!-- 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 starts AC9M8M03. Where it leads: circle problems (next lesson), cylinders (Year 9) and sectors. -->