# Circles

## 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 = 3.2So, using \pi on our calculator, we get

\text{Area }=\pi \times 3.2^2=32.169...=32.2\text{ cm}^2**Example 2: Finding the Circumference**

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

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 8.4mm.

**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.

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

**[2 marks]**

**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]**

The formula for area is

\text{Area }=\pi r^2

In this case, we have \text{area}=200 and r=x. 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)}

**Question 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.

**[2 marks]**

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

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