# Circles

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## 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.

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## 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.
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## 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).

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@mmerevise

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## 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

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

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

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## Circles 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.

Gold Standard Education

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

Gold Standard Education

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

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

Gold Standard Education

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

$\dfrac{120}{\pi} = x$

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