Circles

What you will learn

• Name the parts of a circle.
• Explain what $\pi$ is.
• Find circumference, area, arc length and sector area.

Before you start: you should know how to find the area of a rectangle and how to multiply decimals. See Area and Perimeter.

The idea

A circle is the set of all points at the same distance from a centre point. That fixed distance is the radius ($r$). The diameter ($d$) goes all the way across through the centre, so $d = 2r$.

Parts of a circle

• Circumference: the distance around the outside.
• Chord: a straight line joining two points on the circle. The diameter is the longest chord.
• Arc: part of the circumference.
• Sector: a “pizza slice” bounded by two radii and an arc.
• Tangent: a straight line that touches the circle at exactly one point. It is always perpendicular to the radius at that point.

What pi is

Measure the circumference of any circle, such as a coin, a plate or a bicycle wheel, and divide by its diameter. You always get about $3.14$. This constant ratio is called pi, written $\pi$:

$\pi = \frac{C}{d} \approx 3.14159\ldots$

Its decimal digits never end and never repeat. For most calculations $3.14$ or the calculator’s $\pi$ button is fine.

Why the area is $\pi r^2$

Cut a circle into many thin sectors and rearrange them alternately point-up and point-down. They form a shape close to a rectangle whose height is $r$ and whose length is half the circumference, $\pi r$. Its area is $\pi r \times r = \pi r^2$, and the approximation gets better the thinner the slices.

Sectors and arcs

A sector with angle $\theta$ is the fraction $\theta \div 360$ of the whole circle, so its area and arc length are that same fraction of $\pi r^2$ and $2\pi r$.

Worked examples

Example 1: Circumference and area

A circular pond has a diameter of $8$ m. Find its circumference and area to 1 decimal place.

1. Radius is half the diameter: $r = 4$ m.
2. Circumference: $C = \pi d = \pi \times 8 \approx 25.1$ m.
3. Area: $A = \pi r^2 = \pi \times 16 \approx 50.3 \text{ m}^2$.

Answer: $C \approx 25.1$ m, $A \approx 50.3 \text{ m}^2$

Example 2: Working backwards

A circular table top has an area of $1.13 \text{ m}^2$. What is its radius?

1. Start from $\pi r^2 = 1.13$.
2. Divide by $\pi$: $r^2 = 1.13 \div \pi \approx 0.3597$.
3. Square root: $r \approx 0.60$ m.

Answer: about $60$ cm

Example 3: A sector

A pizza of radius $15$ cm is cut into $8$ equal slices. Find the area of one slice and the length of its crust.

1. Each slice has angle $360^\circ \div 8 = 45^\circ$, which is $\frac{1}{8}$ of the circle.
2. Area: $\frac{1}{8} \times \pi \times 15^2 = \frac{225\pi}{8} \approx 88.4 \text{ cm}^2$.
3. Crust (arc): $\frac{1}{8} \times 2\pi \times 15 = \frac{30\pi}{8} \approx 11.8$ cm.

Answer: about $88.4 \text{ cm}^2$ and $11.8$ cm

Tip: check whether a question gives the radius or the diameter. Using the diameter in $\pi r^2$ makes the area four times too big.

Why it matters

Wheels, gears, pipes, clocks, plates and planets are all circular or nearly so. A car’s speedometer works out speed from how many times the wheels turn, using circumference. Engineers calculate pipe cross-sections with $\pi r^2$ to know how much water can flow through.