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# Circumference

Area, volume, circles and Pythagoras · Circles

<!-- Speaker note: 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 (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** because C ÷ d = π, the circumference is π × diameter: C = πd, or C = 2πr from the radius. Going backwards, d = C ÷ π.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> 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.

<!-- Speaker note: The last lesson found that $C \div d = \pi$ for every circle. Turned round, that is a formula for the circumference: $$C = \pi d \qquad \text{or, since } d = 2r, \qquad C = 2\pi r$$ Use $\pi \approx 3.14$, as the questions say. Going backwards, from a circumference to the diameter, divide by π: $d = C \div \pi$. -->

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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 5 cm. About how long is its circumference?**

- 10 cm
- 15.7 cm
- 31.4 cm
- 78.5 cm

<!-- Speaker note: What happens: 31.4 cm. The diameter is 10 cm, and π × 10 ≈ 31.4. 15.7 uses the radius as the diameter; 10 cm is the diameter itself. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 15.7 used the radius as the diameter. -->

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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%2Fcircumference%2Fstep-2&code=widget%3A+circle%0Aslider+r%3A+0.5%2C+4%2C+1%2C+0.5%0Aroll%3A+on%0Ashow%3A+none%0Aunits%3A+cm%0A&fresh=1}}

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

<!-- Speaker note: Predict the circumference from the diameter before you roll. -->

<!-- Speaker note: Teacher: ask students to predict before rolling, then compare. -->

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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?

* 31.4 cm. The diameter is 10 cm, and π × 10 ≈ 31.4. 15.7 uses the radius as the diameter; 10 cm is the diameter itself.

<!-- 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>

## Use the diameter

Here is the formula forwards and backwards.

Now try one: diameter 8 cm.

$$C = \pi d = 3.14 \times 8 = 25.12 \text{ cm}$$

1) $C = 2\pi r = 2 \times 3.14 \times 30 = 188.4 \text{ cm}$ — From the radius: double it first (2r is the diameter).
2) $d = C \div \pi = 18.84 \div 3.14 = 6 \text{ m}$ — Backwards: divide by π.
3) $C = 10\pi \text{ cm}$ — Exact: leave π in the answer (for d = 10). 10π means 10 × π, about 31.4.

<!-- 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 C = d = 3.14 × 8 = 25.12 cm and say "From the diameter: multiply by π." Line 1: write C = 2 r = 2 × 3.14 × 30 = 188.4 cm and say "From the radius: double it first (2r is the diameter)." Line 2: write d = C ÷ = 18.84 ÷ 3.14 = 6 m and say "Backwards: divide by π." Line 3: write C = 10 cm and say "Exact: leave π in the answer (for d = 10). 10π means 10 × π, about 31.4." -->

<!-- Speaker note: Teacher: share two routes for the wheel: C = π × 60, or C = 2 × π × 30. -->

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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>

## Use the diameter

**A circle has diameter 8 cm. Using π ≈ 3.14, what is its circumference, in cm?**

* Answer: 25.12

<!-- Speaker note: 3.14 × 8 = 25.12 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

**Mei finds how far a wheel rolls in one turn. Click the line where her working goes wrong.**

```
 1  A wheel has radius 30 cm. How far does it roll in one turn?
 2  C = π × d
 3  = 3.14 × 30
 4  = 94.2 cm
```

* Answer: line 3

<!-- Speaker note: Line 3: 30 cm is the radius, but C = π × d needs the diameter, 60 cm. 3.14 × 60 = 188.4 cm (or C = 2πr = 2 × 3.14 × 30). -->

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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 circular garden bed has circumference 18.84 m. Using π ≈ 3.14, find its diameter. Write your steps.**

* Answer: $C = \pi d$, so $18.84 = 3.14 \times d$ (the circumference is π times the diameter); $d = 18.84 \div 3.14 = 6$ m (divide by π to undo multiplying by π)

<!-- Speaker note: 18.84 ÷ 3.14 = 6 m. -->

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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>

## One change at a time



- **Circumference 25:** Each task changes one thing.
- **With 22/7:** 
- **Exact:** 
- **Backwards: the radius:** 
- **A semicircle:** 

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

<!-- Speaker note: What this step is for: 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. Hints, in order: 25 ÷ 3.14 is about 8: that is the diameter. / Radius 4. For students who need more 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). For early finishers: 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? 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. -->

<!-- Speaker note: Teacher: for the semicircle, ask students to trace the whole outline with a finger. -->

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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

**Diameter 12 cm. Circumference, in cm (π ≈ 3.14)?**

* Answer: 37.68

<!-- Speaker note: 3.14 × 12 = 37.68 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>

## From the radius

**Radius 3 cm. Circumference, in cm (π ≈ 3.14)?**

* Answer: 18.84

<!-- Speaker note: 2 × 3.14 × 3 = 18.84 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>

## A bike ride

**A bike wheel has diameter 70 cm. How far does the bike go in 100 turns, in metres (π ≈ 3.14)?**

* Answer: 219.8

<!-- Speaker note: One turn: 3.14 × 70 = 219.8 cm. 100 turns: 21 980 cm = 219.8 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>

## Review: area

**Review: a triangle has base 10 cm and height 6 cm. Its area, in cm²?**

* Answer: 30

<!-- Speaker note: ½ × 10 × 6 = 30 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>

## Make: a trundle wheel

Then work out how many clicks it makes around a 400 m running track.

<!-- Speaker note: What this step is for: 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. Hints, in order: One turn should roll exactly 1 m = 100 cm. / C = πd, so d = 100 ÷ 3.14. -->

<!-- Speaker note: Teacher: build a cardboard trundle wheel if there is time. -->

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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 circumference of a circle with radius 4 cm. A circle has circumference 12.56 cm: find its diameter. (π ≈ 3.14)**

* 2 × 3.14 × 4 = 25.12 cm. 12.56 ÷ 3.14 = 4 cm.

<!-- 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 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). -->