Circumference, Area and Rolling Wheels: Quick Revision

A brief note on Circumference, Area and Rolling Wheels. Read it before a test. Try each question at the end before you open its answer.

The circle

• Circumference: $C = 2\pi r = \pi d$.
• Area: $A = \pi r^2$.
• $\pi$ is the circumference divided by the diameter. Use $\dfrac{22}{7}$ only when the question says so, and write the exact form with $\pi$ as well.
• Doubling the radius doubles the circumference and makes the area four times as large.
• Change all lengths to one unit first. $1\ \text{m}^2 = 10\,000\ \text{cm}^2$.

The ring

• Area of a ring: $\pi(R^2 - r^2) = \pi(R - r)(R + r)$.
• A path outside a circle adds its width to the radius. A path inside takes it away.
• A chord of the big circle that touches the small circle gives the area of the ring: $\pi \times (\text{half the chord})^2$.

The wheel and the hand

• One turn of a wheel rolls $\pi d$. The distance for $n$ turns is $n\pi d$, and the turns for a road $s$ are $\dfrac{s}{\pi d}$.
• A hand that turns through a fraction of a circle sweeps that fraction of the area. A minute hand turns $\dfrac{m}{60}$ in $m$ minutes. An hour hand turns $\dfrac{m}{720}$.

A worked example

The area of a ring is needed. The outer radius is $10.5\ \text{cm}$ and the inner radius is $7\ \text{cm}$. Take $\pi = \dfrac{22}{7}$.

1. $R^2 - r^2 = 110.25 - 49 = 61.25$.
2. $A = \dfrac{22}{7} \times 61.25 = 192.5\ \text{cm}^2$.

Practice with new numbers

A program to try

The program finds the circumference, the area, a ring and the distance of one turn for a circle of radius $10.5\ \text{cm}$.

Mistakes to avoid

• Squaring the difference of the radii. The ring is $\pi(R^2 - r^2)$.
• Using the diameter as the radius.
• Dividing an area in $\text{cm}^2$ by $100$. A square metre is $10\,000\ \text{cm}^2$.
• Taking a full circle for a hand that turns only part of the way.

Check yourself

1. A circle has a radius of $14\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. What is its circumference?

2. A ring has an outer radius of $5\ \text{cm}$ and an inner radius of $3\ \text{cm}$. Write its area in terms of $\pi$.

3. A wheel has a circumference of $2.5\ \text{m}$. How far does it roll in $40$ turns?

4. A minute hand turns for $15$ minutes. What fraction of the circle does it sweep?