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Parts of a circle Circumference of a circle Solving equations Sine rule Cosine rule Rounding numbers Square numbers and square rootsThis topic is relevant for:

Here we will learn about the arc of a circle, including how to find the arc length and work backwards given the arc length to solve problems.

There are also arc 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.

An **arc of a circle** is a portion of the circumference of a circle.

In GCSE mathematics the arc of a circle can be used to define angles and lengths.

The arc of a circle would be the curved edge of a **sector **or **segment**.

There are **circle theorems** that are defined using arcs of the circle.

For example, \theta is the angle **subtended** by arc AC.

We can find the length of an arc if we know the angle of the sector.

The arc length is a fraction of the circumference of the circle.

The fraction will be \frac{\theta}{360}, if \theta is the angle of the sector.

The arc length formula is,

\text{Arc length } = \frac{\theta}{360} \times \pi\times d \theta = \text{ angle of the sector} d = \text{ diameter of the circle}Or

\text{Arc length } = \frac{\theta}{360} \times 2\times\pi \times r \theta = \text{ angle of the sector} r = \text{ radius of the circle}In order to calculate the arc length:

**Find the length of the radius.****Find the size of the angle creating the sector.****Substitute the values you know into the formula for the arc length.****Calculate the answer.**

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

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

COMING SOONCalculate the arc length of the sector shown below.

Give your answer to 3 significant figures.

**Find the length of the radius/diameter.**

Radius = 8 \ cm

2**Find the size of the angle creating the sector**.

Angle = 115^{\circ}

3**Substitute the values you know into the formula for the arc length.**

As you know the radius you can use the formula which has ‘r’ as a variable.

\begin{aligned} \text{Arc length } &=\frac{\theta}{360} \times 2\times\pi \times r \\\\ &=\frac{115}{360} \times 2\times\pi \times 8 \\\\ &=\frac{46}{9} \pi \end{aligned}4**Calculate the answer.**

The question asks you to round your answer to 3 significant figures.

\begin{aligned} \text{Arc length } &=\frac{46}{9}\pi \ cm \\\\ &=16.0570...cm \end{aligned}The arc measures =16.1 \ cm .

Calculate the arc length of the sector shown below.

Give your answer to 3 significant figures.

**Find the length of the radius.**

Radius = 4 \ m

Although we can use the radius or diameter as long as we use the correct formula for each.

**Find the size of the angle creating the sector.**

The angle creating the sector is the angle included between the two radii.

To find it in this case we need to use the angle fact, angles around a point add to 360^{\circ}.

360-240=120

The angle creating the sector is 120^{\circ}.

**Substitute the values you know into the formula for the arc length.**

\begin{aligned}
\text { Arc length }&=\frac{\theta}{360} \times 2 \times \pi \times r \\\\
&=\frac{120}{360} \times 2 \times \pi \times 4
\end{aligned}

**Calculate the answer.**

\begin{aligned} &=8.37758041 \\\\ &=8.38 \mathrm{~m} \end{aligned}

Below is a sector of a circle.

The diameter of the circle is 12 \ cm.

Calculate the arc length.

**Find the length of the radius.**

Diameter = 12 \ cm

The radius is half of the diameter, so the radius = 6 \ cm.

Although we can use the radius or diameter as long as we use the correct formula for each.

**Find the size of the angle creating the sector.**

The angle creating the sector is the angle included between the two radii.

To find it in this case we need to use the angle fact, angles around a point add to 360^{\circ}.

360-205=155

The angle creating the sector is 155^{\circ}.

**Substitute the values you know into the formula for the arc length.**

\begin{aligned}
\text { Arc length }&=\frac{\theta}{360} \times 2 \times \pi \times r \\\\
&=\frac{155}{360} \times 2 \times \pi \times 6
\end{aligned}

**Calculate the answer.**

\begin{aligned}
&=16.23156204 \\\\
&=16.2 \mathrm{~cm}
\end{aligned}

**Finding the length of the circumference not the length of the arc**

Remember to find the fraction of the circle that makes the arc not just the circumference of the whole circle.

For example,

Use \frac{\theta}{360} \times \pi\times d

or \frac{\theta}{360} \times 2\times\pi \times r .

Not \pi\times d

or 2\times\pi \times r.

**Incorrect angle unit set on calculator**

At GCSE all angles are measured in degrees. Make sure that your calculator has a small ‘d’ for degrees at the top of the screen rather than an ‘r’ for radians – these are not used until A Level.

1. What part of a circle does the red line show?

Segment

Major arc

Minor arc

Sector

Part of the circumference is called an arc. The red line is more than half of the circumference, so it is a major arc.

2. The diagram shows a semicircle. What is the length of the curved edge, in terms of \pi.

8 \pi \ cm

8\pi+8 \ cm

4\pi \ cm

2 \pi \ cm

The length of the curved edge of a semicircle can be found by calculating \pi d \div 2.

3. Find the length of the arc of the sector. Give your answer to 1 decimal place.

9.4 \ cm

28.3 \ cm

18.8 \ cm

37.7 \ cm

The length of the curved edge of a quarter-circle can be found by calculating \pi d \div 4.

4. Find the length of the arc of the sector. Give your answer to 1 decimal place.

2.4 \ cm

7.3 \ cm

17.1 \ cm

4.9 \ cm

Substituting the values we know into the formula for the arc length,

\begin{aligned} \text{Arc length } &=\frac{\theta}{360} \times 2 \times \pi \times r \\\\ &=\frac{40}{360} \times 2 \times \pi \times 7 \\\\ &=4.886921906 \mathrm{~cm} \end{aligned}

5. Find the length of the arc of the sector. Give your answer to 1 decimal place.

16.8 \ cm

4.2 \ cm

8.4 \ cm

12.6 \ cm

Substituting the values we know into the formula for the arc length, remembering \theta =60^{\circ} because this is an equilateral triangle.

\begin{aligned} \text{Arc length } &=\frac{\theta}{360} \times 2 \times \pi \times r \\\\ &=\frac{60}{360} \times 2 \times \pi \times 8 \\\\ &=8.37758041 \mathrm{~cm} \end{aligned}

6. Find the angle of the sector with an arc length of 10 \pi \ cm and a radius of 12 \ cm.

150^{\circ}

210^{\circ}

60^{\circ}

30^{\circ}

Substituting the values we know into the formula for arc length,

\frac{\theta}{360} \times 2 \times \pi \times 12=10 \pi

Solving for the angle \theta,

\theta=150^{\circ}.

1. A sector has a radius of 15 \ cm and an angle of 120^{\circ}. Find the length of the arc in terms of \pi.

**(2 marks)**

Show answer

Use of \frac{\theta}{360} \times \pi d .

**(1)**

**(1)**

2. The sector has an angle of 50^{\circ} and an arc length of 25 \ cm. Find the radius of the sector in centimetres to the nearest millimetre.

**(3 marks)**

Show answer

25=\frac{50}{360} \times \pi d (or equivalent).

**(1)**

Diameter = 57.29…

**(1)**

Radius = 28.6 \ cm

**(1)**

3. A metal sector with radius 6 \ cm and angle 250^{\circ} is used to form an open cone, as shown.

Calculate the radius, r, of the cone formed from the sector.

**(4 marks)**

Show answer

Use of \frac{250}{360} \times 12 \pi .

**(1)**

Arc length = 26.1799…

**(1)**

**(1)**

**(1)**

You have now learned how to:

- Identify and apply circle definitions and properties, including tangent, arc, sector and segment
- Calculate the length of an arc
- Calculate a property of a circle given the length of an arc

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