Circles

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Circles Revision

Circles

Circles appear everywhere in maths. Mathematicians just can’t get enough of them.

Here, we’re going to introduce a few of the terms used to describe parts of a circle, and then we’re going to look at calculating the area and perimeter/circumference of a circle. The terms we’ll need are shown on the diagram and described in further detail below.

Key Circle Terms

The circumference is the outside edge of the circle.
A diameter is a straight line going straight through the centre of the circle and touching the circumference at each end.
A chord is a straight line joining any two parts of the circumference.
A segment is the area bound by the circumference and a chord.
An arc is a section of the circumference.
A radius (plural radii, pronounced “ray-dee-eye”) is a straight line joining the centre to the circumference.
A sector is the area bound by two radii and an arc – like a pizza slice.
A tangent is a straight line that touches the circumference at a single point.

Area and Circumference of a Circle

Area of a circle $=\textcolor{red}{\pi} \textcolor{blue}{r}^2$

Circumference of a circle $=\textcolor{red}{\pi} \textcolor{green}{d} = 2\textcolor{red}{\pi}\textcolor{blue}{r}$

Where $\textcolor{blue}{r}$ is the radius, $\textcolor{green}{d}$ is the diameter, and $\textcolor{red}{\pi}$ is a very special number with a specific value of $3.14159265...$ ( $3.14$ to $2$ dp).

Example 1: Area of a Circle

Below is a circle with centre $C$ and radius $3.2$ cm.

Find the area of the circle to $1$ dp.

[2 marks]

Formula: $\text{Area }=\pi r^2$ .

We know the radius is $3.2$ , so we have

r = 3.2

So, using $\pi$ on our calculator, we get

\text{Area }=\pi \times 3.2^2=32.169...=32.2\text{ cm}^2

circumference of a circle example question

Example 2: Finding the Circumference

Below is a circle with centre $C$ and radius $12$ cm.

Find the circumference of this circle.

Leave your answer in terms of $\pi$ .

[1 mark]

Formula: $\text{Circumference }=\pi d$

Where $d$ is the diameter.

We know the radius $=12$

So, we must double the radius to get the diameter.

$\text{diameter }=2\times \text{radius}$

$12 \times 2 = 24$

Now we can find the circumference

$\text{circumference }=24\times \pi=24\pi\text{ cm}$

Circles Example Questions

Question 1: Below is a circle with centre $C$ and diameter $8.4$ mm.

area circumference circle question

a) Find the circumference of the circle. Give your answer in terms of $\pi$ .

[1 mark]

b) Find the area of the circle. Give your answer to $3$ sf.

[2 marks]

Remember to state the units of your answers.

The formula for circumference is $\pi d$ , so we get

$\text{circumference }=\pi \times 8.4=\dfrac{42}{5}\pi\text{ mm}$

The circumference is the distance around the outside, so its units are the same as those of the diameter.

b) The formula for area is $\pi r^2$ , so firstly we have to get the radius by halving the diameter:

$r=8.4\div 2=4.2$

Then we get

$\text{area }=\pi \times 4.2^2=55.417...=55.4\text{ mm}^2\text{ (3sf)}$

Area of shapes is always measured in “squared” units. Circles are no exception.

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Question 2: Calculate the area of the circle below with a radius of $5$ cm, giving your answers in terms of $\pi$ .

area circle question

[2 marks]

\text{Area}=\pi r^2 = \pi \times 5^2= 25\pi \text{ cm}^2

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Question 3: Below is a circle with centre C and radius $x\text{ cm}$ . The area of this circle is $200\text{ cm}^2$ . Find the value of $x$ to $1$ dp.

[2 marks]

area circumference circle question unknown radius