 ## What you need to know

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 of a circle. So, the terms we’ll need are shown on the diagram and described in further detail below.

Centre: the single point which lies at the centre of the circle.

Diameter: any straight line that connects two points on the circumference and passes through the centre. Any diameter you draw will be the same length.

Radius: any straight line that connects the centre to a point a on the circumference. Any radius you draw will be the same length.

Circumference: the distance around the outside of the circle, i.e. the perimeter of the circle.

KEY POINT: the radius is always half the length of the diameter, or in other words, the diameter is always double the length of the radius.

Circle Perimeter and Area

So, let’s see about how we calculate the perimeter and area of a circle. We have already learned that another name for the perimeter of a circle is the circumference, so from now we’ll exclusively be using the word circumference to refer to the perimeter of the circle.

The two formulas you need to know are:

$\text{Area of a circle }=\pi r^2$

$\text{Circumference of a circle }=\pi d$

Here, $r$ is the radius, $d$ is the diameter, and $\pi$ is a Greek letter (the English spelling is ‘pi’ and it’s said like ‘pie’) that we use to refer to a very special number.

Take a moment to find $\pi$ on your calculator and notice that if you type it in and press the equals button, you’ll get something like 3.14159265. In fact, $\pi$ is a decimal that goes on forever and never repeats itself (but your calculator screen, sadly, doesn’t go on forever). Get used to $\pi$ as you’ll be using it a lot in circle problems.

Example: Below is a circle with centre C and radius 3.2cm. Find the area of the circle to 1dp.

The formula is $\text{area }=\pi r^2$. We know the radius is 3.2, so we have

$r = 3.2$

So, using $\pi$ on our calculator, we get (to 1dp)

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

Example: Below is a circle with centre C and radius 12cm. Find the circumference of this circle. Leave your answer in terms of $\pi$.

The formula we’ll need is

$\text{circumference }=\pi d$

Where $d$ is the diameter. But hold on, we don’t know the diameter? In the question we’re only given the radius. Fortunately, we have this very useful fact:

The radius is always half the length of the diameter.

In other words, the diameter is double the radius, or as an equation:

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

We can see how this works in the picture shown below. If you extend a radius in the exact opposite direction away from the centre, then you get a diameter which is made of two radii (that’s the fancy plural of ‘radius’).

So, we know the diameter is 24cm. Now we have to address the “leave your answer in terms of $\pi$” part of the question. This simply means leaving your answer in the form

“something” $\times \pi$

It’s how your calculator naturally displays the answer anyway, and the numbers are less awkward. It’s a good way to display an answer! So, we get

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

You may notice that we removed the $\times$ symbol – this works just like algebra does, as in we tend to write $4x$ rather than $4\times x$.

Example: Below is a circle with centre C, a circumference of 120cm and a diameter of $x$ cm. Find the value of $x$ to 3 significant figures.

The thing that’s different about this question is that we have to work backwards to find $x$. We know the formula we need is

$\text{circumference }=\pi d$

We also know that the circumference is 120 and $d$ is, in this case, $x$. So, filling it in those things that we have into the formula, we get

$120=\pi \times x$

Now we have an equation we can solve. We want $x$, so if we divide both sides by $\pi$, we get

$120\div \pi = x$

Put this into a calculator and we get: $x=38.2$ cm, to 3sf.

## Circles Questions

) 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.

Because we’re finding the perimeter, we need to add up all side-lengths. This includes the big curved one and the straight one.

The straight one is made up of two radii, so its length is

$5\times 2=10\text{ cm}$

Now for the curved bit. It’s a semi-circle which is half a circle. Therefore, the curved bit of a semi-circle must be half the total circumference. The formula for circumference is $\pi d$ and given that the radius is 5cm we must have the diameter to be 10cm. So, we get

$\text{curved part }=(\pi\times 10)\div 2=15.707...\text{ cm}$

Therefore, the perimeter is

$10+15.707...=25.707...=26\text{ cm}\text{ (2sf)}$

The formula for area is

$\text{area }=\pi r^2$

In this case, we have $\text{area}=200$ and r=x[\latex]. So, putting these values into the formula above, we get the equation

200 = \pi x^2

We can now rearrange this equation to find $x$. Firstly, divide by $\pi$ to get

$\dfrac{200}{\pi}=x^2$

Then, to find out the value of $x$, square root both sides

$x=\sqrt{\dfrac{200}{\pi}}=7.97...=8.0\text{ cm (1dp)}$