## 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.

## 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}

Where \textcolor{blue}{r} is the radius, \textcolor{green}{d} is the diameter, and \textcolor{red}{\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.

## Example 1

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

Find the area of the circle to 1dp.

**[1 mark]**

The formula is \text{area }=\pi r^2.

We know the radius is 3.2, so we have

r = 3.2So, using \pi on our calculator, we get (to 1dp)

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

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.

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

No we can find the circumference

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

### Example 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</p> <p> </p> <p style="text-align: left;">[latex]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)}

4) 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 dWe 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 xNow we have an equation we can solve. We want x, so if we divide both sides by \pi, we get

120\div \pi = xPut this into a calculator and we get: x=38.2 cm, to 3sf.

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#### (NEW) Circles - The basics Exam style questions - MME

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