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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.

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 $=textcolor{red}{pi} textcolor{green}{d} = 2textcolor{red}{pi}textcolor{blue}{r}$

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

$text{Area }=pi times 3.2^2=32.169…=32.2text{ cm}^2$

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.

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

$12 times 2 = 24$

Now we can find the circumference

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

circumference of a circle example question

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

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

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:

$r=8.4div 2=4.2$

Then we get

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

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]

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

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

[2 marks]

area circumference circle question unknown radius

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

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

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Question 4: Below is a circle with centre $C$ , a circumference of $120$ cm and a diameter of $x$ cm.

Find the value of $x$ to $3$ significant figures.

[2 marks]

area circle question unknown diameter

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.

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Specification Points Covered

Geometry and measures – 9. identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

Geometry and measures – 17. know the formulae: circumference of a circle $2 pi r = pi d$ , area of a circle $= pi r^2$ ; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids