Circles: circumference & area

What you will learn

use $C = 2\pi r$ (or $C = \pi d$ ) to find the circumference,
use $A = \pi r^2$ to find the area of a circle,
solve reverse problems: given circumference or area, find the radius,
combine circles with other shapes (semicircles, sectors, composite circle shapes).

Worked example 0 Real-world example: how far does a bike travel per wheel turn?

A bike wheel has a diameter of $66$ cm. How far does the bike move in one full rotation?

Radius $r = 66 \div 2 = 33$ cm.
Circumference $C = 2\pi r = 2 \times \pi \times 33 = 66\pi \approx 207.3$ cm.
One full rotation moves the bike about $2.07$ m forward.
Bonus: in a $1$ km ride, the wheel turns $\dfrac{1000}{2.073} \approx 483$ times.

Key idea: the circumference formula converts a rotation (something you can see) into a straight-line distance (something you can measure).

1. Recap: parts of a circle

Parts of a circle: centre, radius r, diameter d (through the centre), and circumference C (the whole distance around).

Centre: the point in the middle.
Radius ( $r$ ): from centre to any point on the edge.
Diameter ( $d$ ): straight across through the centre; $d = 2r$ .
Circumference ( $C$ ): the total distance around the circle.
$\pi \approx 3.14159$ (irrational). Common approximations: $3.14$ and $\tfrac{22}{7}$ .

Why C = πd: if you 'unroll' a circle's edge into a straight line, that line is exactly π times the diameter. It takes just over 3 diameters to wrap around.

2. The two formulas

Circle formulas

Circumference

C = 2 \pi r \;=\; \pi d.

Area

A = \pi r^2.

Worked example 1 Circumference

A circle has radius $7$ cm. Find its circumference using $\pi \approx \tfrac{22}{7}$ .

C = 2 \pi r = 2 \times \tfrac{22}{7} \times 7 = 44 \text{ cm}.

Worked example 2 Area

A circle has diameter $20$ cm. Find its area using $\pi \approx 3.14$ .

Radius: $r = \tfrac{20}{2} = 10$ cm.
Area: $A = \pi r^2 = 3.14 \times 100 = 314$ cm $^2$ .

3. Reverse problems

Worked example 3 Find the radius from circumference

A circle has circumference $31.4$ cm. Find its radius (using $\pi \approx 3.14$ ).

r = \dfrac{C}{2\pi} = \dfrac{31.4}{6.28} = 5 \text{ cm}.

Worked example 4 Find the radius from area

A circle has area $154$ cm $^2$ . Find its radius (using $\pi \approx \tfrac{22}{7}$ ).

$A = \pi r^2$ , so $r^2 = \dfrac{A}{\pi} = \dfrac{154}{22/7} = 154 \times \dfrac{7}{22} = 49$ .
$r = \sqrt{49} = 7$ cm.

4. Semicircles and sectors

A semicircle is half a circle. A sector is a pie-slice fraction of a circle.

Semicircle area: $\tfrac{1}{2} \pi r^2$ .
Quarter-circle area: $\tfrac{1}{4} \pi r^2$ .

For the perimeter of a semicircle, remember the straight edge (the diameter): $P = \pi r + 2r$ .

Worked example 5 Semicircle perimeter and area

A semicircle has radius $5$ cm. Find its area and perimeter (use $\pi \approx 3.14$ ).

Area: $\tfrac{1}{2} \times 3.14 \times 25 = 39.25$ cm $^2$ .
Perimeter: $3.14 \times 5 + 2 \times 5 = 15.7 + 10 = 25.7$ cm.

Practice: Year 8 core

Where a value of $\pi$ is not given, use $\pi \approx 3.14$ .

Fluency

Circumference and area

A circle has radius $4$ cm. Find its circumference.
A circle has diameter $10$ m. Find its circumference.
A circle has radius $9$ cm. Find its area.
A circle has diameter $14$ cm. Find its area (use $\pi \approx \tfrac{22}{7}$ ).
A circle has radius $2.5$ m. Find its circumference.
A circle has radius $5$ cm. Find its area.

Fluency

Reverse problems

A circle’s circumference is $62.8$ cm. Find its radius.
A circle’s area is $50.24$ m $^2$ . Find its radius.
A wheel has circumference $1.57$ m. Find its diameter.
A round plate has area $314$ cm $^2$ . Find its diameter.

Fluency

Semicircles and sectors

A semicircle has radius $6$ cm. Find its area and perimeter.
A quarter-circle has radius $4$ m. Find its area.
A half-moon window is a semicircle of diameter $1$ m. Find its area.
A pizza slice is a sector making $\tfrac{1}{8}$ of a pizza of radius $12$ cm. Find its area.

Reasoning

Explain and spot the mistake

Kira writes $C = \pi r$ for a circle’s circumference. Is Kira correct? If not, what should it be?
Without calculating, decide which changes more: doubling the radius doubles or quadruples the area? Justify.
A student writes the area of a circle with radius $5$ cm as $\pi \times 10$ cm $^2$ . Explain the mistake.
Explain why the circumference of any circle divided by its diameter always gives the same number (which we call $\pi$ ).

Problem solving

Real contexts

A round pool has radius $3.5$ m. A fence is to be built around it, $0.5$ m from the edge. How long is the fence?
A pizza has diameter $32$ cm. Find its area, then the area of one slice if cut into $8$ equal pieces.
A circular lawn has diameter $10$ m. A mower cuts $0.5$ m $^2$ per second. How long to mow (seconds, then minutes)?
A running track is $400$ m around, consisting of two straights of $100$ m each and two semicircular ends. Find the radius of the semicircles (use $\pi \approx 3.14$ ).

Challenge

Reasoning

Harder circles

A rectangle $6$ cm by $4$ cm has a semicircle attached to each short side (so the two semicircles form one full circle). Find the total perimeter and area.
Two circles: small has radius $3$ cm, large has radius $5$ cm. Find the area of the ring between them (the annulus).
A running track sector is $\tfrac{1}{4}$ of a circle with radius $50$ m. Find its arc length.
A pizza shop offers a $30$ cm diameter pizza for $14 or a $40$ cm diameter pizza for $22. Which is better value per cm $^2$ ?