# Circles and Circumference

> Radius, diameter and circumference, estimating with "about 3 diameters", and exact working with pi as 22/7 or 3.14.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/circumference/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The bike wheel
Zoe wants to know how far her bike travels when the wheel turns once. She puts a chalk mark on the tyre, lines it up with a crack in the path, and pushes the bike forward until the mark comes round again.

She measures the distance: 1.88 m. The wheel is 60 cm across. She notices that 1.88 m is a bit more than **3 times** 60 cm.

Every circle in the world does this. The distance around it is always a little more than 3 times the distance across it. That "little more than 3" is a famous number called **pi**.

**In short:** The circumference is the distance around a circle: $C = \pi d$, where $d$ is the diameter (twice the radius) and $\pi \approx \frac{22}{7}$ or 3.14. A quick estimate is that the circumference is about 3 diameters.

## Before you start

- **Perimeter** is the distance around a shape.
- Multiplying by a fraction: $\frac{22}{7} \times 14 = 22 \times 2 = 44$.
- Multiplying decimals: $3.14 \times 10 = 31.4$.

## Parts of a circle

*Diagram: A circle with centre O, a diameter going right across through the centre, and a radius from the centre to the edge*
*The diameter goes right across through the centre. The radius is half of it.*

| Part | Meaning |
|---|---|
| **Centre** (O) | the middle point |
| **Radius**, $r$ | from the centre to the edge |
| **Diameter**, $d$ | right across, through the centre. $d = 2r$ |
| **Circumference**, $C$ | the distance all the way around: the circle's perimeter |

## Estimating: about 3 diameters

### Key idea: The quick estimate
$$C \approx 3 \times d \quad \text{or} \quad C \approx 6 \times r$$

A circle with diameter 12 cm has a circumference of about 36 cm.

Going backwards: if the circumference is about 42 cm, the radius is about $42 \div 6 = 7$ cm.

## The exact formula

### Key idea: Circumference
$$C = \pi \times d \qquad \text{or} \qquad C = 2 \times \pi \times r$$

$\pi$ (pi) is about $3.14$, or about $\frac{22}{7}$.

$\frac{22}{7} = 3\frac{1}{7}$, which is just a bit more than 3. It is a little more accurate than $3.14$. Use whichever the question tells you.

### Exam tip: Pick the easy pi
If the diameter is a multiple of 7 (7, 14, 21, 28...), $\frac{22}{7}$ cancels nicely. Otherwise use 3.14.

### Worked example: With 22/7
Diameter 14 cm: $C = \frac{22}{7} \times 14 = 22 \times 2 = 44$ cm.

Diameter 21 cm: $C = 22 \times 3 = 66$ cm.

Radius 7 cm: the diameter is 14 cm, so $C = 44$ cm.

### Worked example: With 3.14
Diameter 10 cm: $C = 3.14 \times 10 = 31.4$ cm.

Radius 5 cm: diameter 10 cm, so $C = 31.4$ cm.

Diameter 3 cm: $C = 3.14 \times 3 = 9.42$ cm.

Radius 70 cm: diameter 140 cm, so $C = 3.14 \times 140 = 439.6$ cm.

### Watch out: Radius or diameter?
Read carefully. If the question gives the **radius**, double it before using $C = \pi d$. Forgetting to double gives an answer exactly half of the right one.

### Challenge: Selective level: working backwards
A circle has a circumference of 88 cm. Using $\pi = \frac{22}{7}$: $\frac{22}{7} \times d = 88$, so $d = 88 \times \frac{7}{22} = 28$ cm and the radius is 14 cm.

Zoe's wheel has diameter 60 cm. In one turn it rolls $3.14 \times 60 = 188.4$ cm. To roll 1 km, which is $100\,000$ cm, it turns about $100\,000 \div 188.4 \approx 531$ times.

**Mind map**

```text
Circles
  Parts
    Centre O
    Radius r
    Diameter d = 2r
    Circumference C
  Estimate
    C ≈ 3d
    C ≈ 6r
  Exact
    C = πd
    π ≈ 22/7 or 3.14
  Traps
    Radius given? Double it
    Use the π you are told
```

## Quick summary

- The radius goes from the centre to the edge. The diameter goes right across, and $d = 2r$.
- Circumference is the distance around a circle.
- Estimate: $C$ is about 3 diameters or 6 radii.
- Exact: $C = \pi d$, with $\pi \approx \frac{22}{7}$ or $3.14$.
- Use $\frac{22}{7}$ when the diameter is a multiple of 7.
- If you are given the radius, double it first.

## Quiz

Try these questions from the chapter quiz.

1. A circle has radius 9 cm. What is its diameter?
   - A. 4.5 cm
   - B. 18 cm
   - C. 27 cm
   - D. 81 cm
   - Answer: B. 18 cm. The diameter is twice the radius.

2. Estimate the circumference of a circle with diameter 12 cm.
   - A. about 12 cm
   - B. about 24 cm
   - C. about 36 cm
   - D. about 144 cm
   - Answer: C. about 36 cm. About 3 diameters: $3 \times 12 = 36$.

3. Using $\pi = \frac{22}{7}$, find the circumference of a circle with diameter 7 cm.
   - A. 22 cm
   - B. 44 cm
   - C. 49 cm
   - D. 11 cm
   - Answer: A. 22 cm. $\frac{22}{7} \times 7 = 22$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/circumference/#quiz