GCSE Maths Geometry and Measure 2D Shapes CSA

Equation Of A Circle

Here we will learn about the **equation of a circle** including how to recognise the equation of a circle, form an equation of a circle given its radius and centre, use the equation of a circle to find its centre and radius, and solve problems.

There are also equation of a circle worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is the equation of a circle?

The **equation of a circle** is x^2 + y^2 = r^2 , where r represents the radius (with a centre at 0,0 ).

The definition of a circle is a set of all points on a plane that are a fixed distance from a centre. That distance is called the radius.

For GCSE, you need to be able to recognise and use the equation of a circle with centre at the origin so consider the drawing below of a circle on a set of axes:

Now consider a right angled triangle created when the radius of the circle is the hypotenuse of the triangle (see the below figures):

- The horizontal line is the distance to the x coordinate.
- The vertical line is the distance to the y coordinate.
- The hypotenuse is the distance of the radius.

You can now apply Pythagoras’ theorem to the above:

a^2+b^2=c^2

a = x coordinate

b = y coordinate

c = radius

Therefore the general form of the equation of a circle centred at (0,0) is:

x^2+y^2=r^2

E.g.

Draw circle with equation x^2+y^2=9

The circle has a centre at (0,0) .

9 represents r^2 , so the radius r is given by 3 .

### What is the equation of a circle?

## How to use the equation of a circle

In order to solve problems involving the equation of a circle:

**Write the general equation of a circle.****State any variables you know.****Substitute any values you know into the equation.****Use the information you have to solve the problem.****Clearly state the answer.**

### How to use the equation of a circle

### Equation of a circle worksheet

Get your free equation of a circle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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### Equation of a circle worksheet

Get your free equation of a circle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD NOW

### Related lessons on circles, sectors and arcs

**Equation of a circle** is part of our series of lessons to support revision on **circles, sectors and arcs**. You may find it helpful to start with the main circles, sectors and arcs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

- Circles, sectors and arcs
- Segment of a circle
- Area of a segment
- Sector of a circle
- Arc of a circle
- Parts of a circle

## Equation of a circle examples

### Example 1: writing the equation of a circle with a centre at the origin

What is the equation of a circle with a radius of 3 and a centre at the origin

**Write the general equation of a circle**.

x^2+y^2=r^2

2**State any variables you know.**

Radius = 3

3**Substitute any values you know into the equation.**

x^2+y^2=3^2

4**Use the information you have to solve the problem.**

\[\begin{aligned}&x^{2}+y^{2}=3^{2} \quad \quad \quad \text{Simplify the equation by squaring the radius} \\&x^{2}+y^{2}=9\end{aligned}\]

5**Clearly state the answer.**

Equation of the circle is :

x^2+y^2=9

### Example 2: writing the equation of a circle with a centre at the origin

What is the equation of a circle with a radius of 1.5 and a centre at the origin.

**Write the general equation of a circle.**

x^2+y^2=r^2

**State any variables you know.**

Radius = 1.5

**Substitute any values you know into the equation.**

x^2+y^2=1.5^2

**Use the information you have to solve the problem.**

\[\begin{aligned}&x^{2}+y^{2}=1.5^{2} \quad \quad \quad \text {Simplify the equation by squaring the radius} \\&x^{2}+y^{2}=2.25\end{aligned}\]

**Clearly state the answer.**

Equation of the circle is:

x^2+y^2=2.25

### Example 3: writing the equation of a circle with a centre at the origin

What is the equation of a circle with a radius of \sqrt5 and a centre at the origin.

**Write the general equation of a circle.**

x^2+y^2=r^2

**State any variables you know.**

Radius = \sqrt5

**Substitute any values you know into the equation.**

x^2+y^2=\sqrt5^2

**Use the information you have to solve the problem.**

\[\begin{aligned}&x^{2}+y^{2}=(\sqrt{5})^{2} \quad \quad \quad \text {Simplify the equation by squaring the radius } \\&x^{2}+y^{2}=5\end{aligned}\]

**Clearly state the answer.**

Equation of the circle is:

x^2+y^2=5

### Example 4: finding the radius of a circle given its equation

What is the radius of the circle with the equation

x^2+y^2=4

**Write the general equation of a circle.**

x^2+y^2=r^2

**State any variables you know.**

Radius is unknown

Equation of the circle is given as x^2+y^2=4

**Substitute any values you know into the equation.**

We do not know any variables so we are unable to substitute here.

**Use the information you have to solve the problem.**

We that the radius squared is equal to 4 , so

\[\begin{aligned}r^{2}&=4 \\r&=\sqrt{4} \\r&=2 \quad \text {You only use the positive value as radius is measure of distance}\end{aligned}\]

**Clearly state the answer.**

The radius is 2.

### Example 5: finding the radius of a circle given its equation

What is the radius of the circle with the equation

x^2+y^2=15

Give you answer to 2 decimal places.

**Write the general equation of a circle.**

x^2+y^2=r^2

**State any variables you know.**

Radius is unknown. Equation of the circle is given as x^2+y^2=15

**Substitute any values you know into the equation.**

We do not know any variables so we are unable to substitute here.

**Use the information you have to solve the problem required.**

We know the equation of the circle is given in the question, therefore, we can create an equation for the radius:

\[\begin{aligned}r^{2}&=15 \\r&=\sqrt{15} \\r&=3.87298 \quad \text {You only use the positive value as radius is measure of distance}\end{aligned}\]

**Clearly state the answer.**

The question asks for the answer to be given to 2 decimal places, therefore: r = 3.87

### Example 6: finding the radius of a circle given its equation

What is the radius of the following circle?

2x^2+2y^2=100

Give your answer to 1 decimal place.

**Write the general equation of a circle.**

x^2+y^2=r^2

**State any variables you know.**

Radius is unknown. Equation of the circle is given as 2x^2+2y^2=100

**Substitute any values you know into the equation.**

We do not know any variables so we are unable to substitute here.

**Use the information you have to solve the problem required.**

We need to rearrange the equation so it is in the form:

x^2+y^2=r^2

Therefore,

\[\begin{aligned}2 x^{2}+2 y^{2}&=100 \quad \quad \quad \text { Divide all terms by 2 } \\x^{2}+y^{2}&=50\end{aligned}\]

And so:

\[\begin{aligned}&r^{2}=50 \\&r=\sqrt{50} \\&r=7.071 \quad\end{aligned}\]

**Clearly state the answer.**

The question asks for the answer to be given to 1 decimal places, therefore: r=7.1

### Common misconceptions

**The radius squared**

Remember in the equation the radius is shown as a squared value. To find the radius you need to square root this value.

**The radius and negatives**

The radius cannot be negative because as the radius is a length it must always be a positive value.

**Equation in correct form**

Remember in order to apply the equation of a circle to a question it must be in the form: x^2+y^2=r^2

### Practice equation of a circle questions

1. What is the equation of a circle with a radius of 4 and a centre at the origin?

x^2+y^2=16

x^2+y^2=4

x^2+y^2=2

16

\begin{aligned}x^2+y^2=r^2\\x^2+y^2=4^2\\x^2+y^2=16\end{aligned}

2. What is the equation of a circle with a radius of 6 and a centre at the origin?

36

x^2+y^2=6

x^2+y^2=12

x^2+y^2=36

\begin{aligned}x^2+y^2=r^2\\x^2+y^2=6^2\\x^2+y^2=36\end{aligned}

3. What is the equation of a circle with a radius of \sqrt3 and a centre at the origin?

x^2+y^2=1.73

x^2+y^2=3

3

x^2+y^2=9

\begin{aligned}x^2+y^2=r^2\\x^2+y^2=\sqrt3^2\\x^2+y^2=3\end{aligned}

4. What is the radius of the circle with the equation x^2+y^2=100

10000

10

100

x^2+y^2=10

\begin{aligned}r^{2}=100\\r=\sqrt{100}\\r=10\end{aligned}

5. What is the radius of the circle with the equation (to two decimal places):

x^2+y^2=20

20

4.47

400

x^2+y^2=10

\begin{aligned}r^{2}=20 \\r=\sqrt{20} \\r=4.47\end{aligned}

6. What is the radius of the circle with the equation (to two decimal places):

4x^2+4y^2=64

64

16

4

2

\begin{aligned}4x^2+4y^2=64\\x^{2}+y^{2}=16\end{aligned}

Therefore,

\begin{aligned}r^{2}=16 \\r=\sqrt{16} \\r=4\end{aligned}

### Equation of a circle GCSE questions

1. What is the equation of the circle with centre (0, 0) and radius of 2\sqrt3 units?

**(2 marks)**

Show answer

2\sqrt3 being squared at any point, or 12 seen

**(1)**

x^2+y^2=12

**(1)**

2. What is the equation of the circle with centre (0, 0) and radius of \frac{2}{3} units?

**(2 marks)**

Show answer

\frac{2}{3} being squared at any point, or \frac{4}{9} oe seen

**(1)**

x^2+y^2=\frac{4}{9} oe

**(1)**

3. Find the radius of the circle with the following equation.

Give your answer in the form a\sqrt b

**(3 marks)**

Show answer

Divide equation by 5 or x^2+y^2=40 seen

**(1)**

\sqrt40

**(1)**

2\sqrt10

**(1)**

## Learning checklist

You have now learned how to:

- Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference
- Recognise and use the general equation of a circle with centre at the origin

## The next lessons are

- How to work out perimeter
- Angle rules
- Angles in polygons

## Still stuck?

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