Circles - Circumference and Area: Edexcel GCSE Foundation maths revision notes

Circles in the exam

In the exam you may be asked to name a part of a circle, such as the radius or the diameter, for 1 mark. Working out a circumference or an area is usually worth 2 marks.

Some questions say "Give your answer in terms of \pi". Others ask you to round, for example correct to 1 decimal place. Paper 1 has no calculator, so there you usually leave \pi in your answer. If a question on Paper 1 asks for an estimate, use 3 for \pi.

Harder questions, for 4 or 5 marks, use half or quarter circles, or a circle joined to or cut out of another shape. They often end with a cost, or a number of bags or tins to buy.

Since 2022 a formulae sheet has been given in the exam, with both circle formulae on it, but learn the formulae anyway. You still need to know which one to use, and whether it needs the radius or the diameter.

Parts of a circle

The centre is the point in the middle. The radius is a straight line from the centre to the edge. The diameter is a straight line from edge to edge through the centre, so it is two radii long:

d=2r

where d is the diameter and r is the radius.

The circumference is the distance all the way round the edge of the circle. A radius or a diameter can point in any direction, not only across or up and down.

A chord is a straight line from edge to edge that does not go through the centre. A tangent is a straight line outside the circle that touches it at just one point.

An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc, like a slice of pizza. A segment is the part of a circle cut off by a chord.

Working out the circumference

The circumference of any circle is a little more than 3 times its diameter. The exact number is called pi, written \pi:

C=\pi d

where C is the circumference and d is the diameter.

Because d=2r, the same formula can be written C=2\pi r. Use whichever matches the length you are given. Circumference is a length, so it is measured in units such as cm or m.

\pi=3.14159\ldots and its digits never end. Use the \pi key on your calculator and round at the end. For an answer in terms of \pi, leave \pi in it: a circle with diameter 5 cm has circumference 5\pi cm.

Worked example 1: Circumference from a diameter

A round clock face has a diameter of 26 cm. Work out the circumference of the clock face. Give your answer correct to 1 decimal place.

The circumference is the distance round the edge. The 26 cm line goes right across through the centre, so it is the diameter. Use C=\pi d
C=\pi\times26
Your turn. Work out \pi\times26 on your calculator. Give your answer correct to 1 decimal place.

Your answer

\pi\times26=81.681\ldots, which is 81.7 to 1 decimal place.
Check: \pi is a bit more than 3, and 3\times26=78. So an answer a bit more than 78 is sensible.
Answer: 81.7 cm

Working out the area

The area of a circle is the space inside it:

A=\pi r^2

where A is the area and r is the radius. r^2 means r\times r, so a radius of 3 cm gives \pi\times3\times3=9\pi\text{ cm}^2

The formula uses the radius. If you are given the diameter, halve it first. Putting the diameter in by mistake makes the area four times too big.

For an answer in terms of \pi, work out r^2 and write \pi after it. For a decimal answer, use the \pi key. Area is measured in square units, such as \text{cm}^2 or \text{m}^2

Worked example 2: Area in terms of pi

A circular coaster has a diameter of 14 cm. Work out the area of the coaster. Give your answer in terms of \pi

The area formula A=\pi r^2 needs the radius. The 14 cm line goes through the centre, so it is a diameter.
Your turn. What is the radius of the coaster, in cm?

Your answer

r=14\div2=7 cm
A=\pi\times7^2=\pi\times7\times7
7\times7=49, so A=49\pi. Using 14 in place of 7 would give 196\pi, which is four times too big.
Answer: 49\pi\text{ cm}^2

Semicircles and quarter circles

A semicircle is half a circle, so its area is \pi r^2\div2. A quarter circle is a quarter of a circle, so its area is \pi r^2\div4

The perimeter is the whole way round the outside of a shape. For a semicircle, that is half the circumference plus the straight edge, which is a diameter. For a quarter circle, it is a quarter of the circumference plus two radii.

This quarter circle has radius 4 cm. Its area is \pi\times4^2\div4=4\pi\text{ cm}^2. Its curved edge is \pi\times8\div4=2\pi cm, so its perimeter is 2\pi+4+4=2\pi+8 cm.

Worked example 3: Perimeter of a semicircle

A flower bed is a semicircle with a diameter of 6 m. Work out the perimeter of the flower bed. Give your answer correct to 1 decimal place.

The perimeter is the curved edge plus the straight edge. The straight edge is the diameter, 6 m.
A full circle with diameter 6 m has circumference \pi\times6=18.849\ldots m
The curved edge is half of this: 18.849\ldots\div2=9.424\ldots m
Your turn. Add the straight edge. Work out 9.424\ldots+6, correct to 1 decimal place.

Your answer

9.424\ldots+6=15.424\ldots, which is 15.4 to 1 decimal place.
Answer: 15.4 m

Working backwards to a length

Sometimes you are given the circumference and asked for the diameter or the radius. Undo the formula: since C=\pi d, divide the circumference by \pi to get the diameter. Halve that if you need the radius.

Given the area, divide by \pi to get r^2, then find the square root. A circle with area 36\pi\text{ cm}^2 has r^2=36, so its radius is 6 cm.

Worked example 4: Diameter from a circumference

A measuring wheel has a circumference of 100 cm. Work out the diameter of the wheel. Give your answer correct to the nearest centimetre.

Start from C=\pi d and put in the circumference: 100=\pi\times d
To undo multiplying by \pi, divide by \pi: d=100\div\pi
Your turn. Work out 100\div\pi, correct to the nearest whole number.

Your answer

100\div\pi=31.830\ldots, which is 32 to the nearest whole number.
Check: \pi\times32=100.53\ldots, very close to 100
Answer: 32 cm

Circles in other shapes

Harder questions join a circle, a semicircle or a quarter circle to a rectangle or a triangle. Split the shape into parts you know, work out the area of each part, then add them.

When a circle is cut out of a shape, or a round pond sits in a lawn, subtract: the area left is the area of the whole shape take away the area of the circle.

Keep the full calculator value until the end. If you then need bags, tins or boxes, round up: 2.1 tins means you must buy 3 tins, because 2 tins are not enough.

Worked example 5: A rectangle with a semicircle on top

A window is a rectangle with a semicircle on top. The rectangle is 0.8 m wide and 1.2 m tall. Glass costs £45 per square metre. Work out the cost of the glass for the window.

Split the window into the rectangle and the semicircle. Work out each area, then add them.
Rectangle: 0.8\times1.2=0.96\text{ m}^2
The semicircle sits on the top edge, so its diameter is 0.8 m. Its radius is 0.8\div2=0.4 m
Your turn. Work out the area of the semicircle in \text{m}^2, correct to 2 decimal places.

Your answer

\pi\times0.4^2\div2=0.251\ldots\text{ m}^2, which is 0.25 to 2 decimal places.
Total area: 0.96+0.251\ldots=1.211\ldots\text{ m}^2. Cost: 1.211\ldots\times45=54.509\ldots
Answer: \pounds 54.51

Worked example 6: A shape with a circle cut out

A sign is a rectangle 3 m long and 2 m high. A circular hole with a diameter of 1.2 m is cut out of it. One side of the sign is painted. One tin of paint covers 2\text{ m}^2. How many tins of paint are needed?

The painted part is the rectangle with the circle taken away.
Rectangle: 3\times2=6\text{ m}^2. The radius of the hole is 1.2\div2=0.6 m
Hole: \pi\times0.6^2=1.130\ldots\text{ m}^2
Your turn. Work out the area to be painted, 6-1.130\ldots, correct to 2 decimal places.

Your answer

6-1.130\ldots=4.869\ldots\text{ m}^2, which is 4.87 to 2 decimal places.
Tins: 4.869\ldots\div2=2.434\ldots. So 2 tins are not enough: round up.
Answer: 3 tins

Common mistakes

Using the diameter as the radius. Halve the diameter before you put it into \pi r^2
Mixing up the two formulae. Round the edge is the circumference, \pi d; the space inside is the area, \pi r^2
Squaring the wrong thing. Only the radius is squared: \pi\times8^2=64\pi, not \pi^2\times8
Forgetting to halve or quarter. Find the whole circle's area, then divide by 2 for a semicircle or by 4 for a quarter circle.
Leaving out the straight edges. A semicircle's perimeter includes its diameter; a quarter circle's includes two radii.
Stopping too soon in a long question. Subtract a hole from the whole shape, and round up the number of bags or tins.