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17.3 The area and circumference of a circle

17.3 The area and circumference of a circle

The circumference of a circle is the path of points that are all the same distance from the centre.
You can work out the length of the circumference (perimeter) of a circle, with the formulae:
C = πd where C is the circumference and d is the diameter of the circle.
You can also use:

C = 2πr where C is the circumference and r is the radius of the circle.
Remember that diameter = 2 × radius, which is why there
are two versions of the formula. From the rules of algebra:
C = πd means C = π × d
The one you use depends on the information you are given.
C = 2πr means C = 2 × π × r
r When you are told the diameter, use C = πd.
r When you are told the radius, use C = 2πr.

You can work out the area of a circle, with the formula:
From the rules of algebra:
2
A = πr where A is the area and r is the radius of the circle.
2
2
A = πr means A = π × r
2
In the area formula you must use the radius. r = r × r
A common mistake is to work
When you are told the diameter, calculate the
out r as r × 2 instead of r × r.
2
radius (radius = diameter ÷ 2), then use the formula.
Worked example 17.3

Work out: i the circumference ii the area of a circle with
a radius 4 cm b diameter 3 m.
Use π = 3.14 and round your answers correct to 1 decimal place (1 d.p.).

a i C = 2πr You are given the radius, so write down the formula you are going to use.
= 2 × π × 4 Substitute r = 4 into the formula. You don’t need to write 3.14 for π.
= 25.12 Work out the answer.
= 25.1 cm Round your answer correct to 1 d.p. and remember to write the units, cm.
2
ii A = πr Start by writing down the formula you are going to use.
= π × 4 2 Substitute r = 4 into the formula. You don’t need to write 3.14 for π.
2
= π × 16 Work out 4 first.
= 50.24 Work out the answer.
2
2
= 50.2 cm Round your answer correct to 1 d.p. and remember to write the units, cm .
b i C = πd You are told the diameter so write down the formula you are going to use.
= π × 3 Substitute d = 3 into the formula.
= 9.42 Work out the answer.
= 9.4 m Round your answer correct to 1 d.p. and remember to write the units, m.
2
ii A = πr Start by writing down the formula you are going to use.
r = d ÷ 2 You know the diameter, but you need the radius, so work out the radius first.
= 3 ÷ 2 = 1.5
A = π × 1.5 2 Substitute r = 1.5 into the formula.
2
= π × 2.25 Work out 1.5 first.
= 7.065 Work out the answer.
2
2
= 7.1 m Round your answer correct to 1 d.p. and remember to write the units, m .
17 Area, perimeter and volume 169