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Coordinates

Distance between two points

Parts of a circle Equations Quadratic equations Expanding brackets Solving equations Surds SubstitutionThis topic is relevant for:

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.

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 .

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.**

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

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

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

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

2**State any variables you know.**

Radius = 3

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

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=9What 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=2.25What 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=5What 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.

What is the radius of the circle with the equation

x^2+y^2=15Give 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

What is the radius of the following circle?

2x^2+2y^2=100Give 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^2Therefore,

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

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

**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

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}

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)**

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

- Circle theorems
- Vectors
- Standard form
- Perimeter of a sector

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