# Arc Length and Sector Area

GCSELevel 4-5Level 6-7Cambridge iGCSE

## Arc Length and Sector Area

An arc is part of the circumference of a circle, and you will need to know how to calculate the length of an arc based on the angle at the centre of the circle.

A sector a section of a circle, with the edges being two radii and an arc. You will need to know how to calculate the area of a sector.

## Arc Length

Arc length can be calculated using the following equation:

Length of Arc $= \dfrac{x}{360}\times$ Circumference of full Circle

Remember:

Circumference of a circle $=\pi d$

Where $d$ is the diameter

Level 4-5GCSECambridge iGCSE

## Sector Area

Sector area can be calculated using the following equation:

Area of Sector $= \dfrac{x}{360}\times$ Area of full Circle

Remember:

Area of a circle $= \pi r^2$

Where $r$ is the radius

Level 4-5GCSECambridge iGCSE

## Example 1: Arc Length in terms of $\pi$

Find the length of the following arc, giving your answer in its exact form.

[3 marks]

When a question asks for an answer in its exact form, we can give the answer in terms of $\pi$.

Firstly, we need to find the overall circumference of the circle that the arc is part of.

The equation for the circumference of a circle is:

Circumference $= \pi d$

As the radius is $8\text{ cm}$, the diameter is $16\text{ cm}$, so the circumference is $16\pi$

We can now work out the length of the arc:

Length of Arc $= \dfrac{x}{360}\times$ Circumference of full Circle

Length of Arc $= \dfrac{45}{360}\times16\pi$

Length of Arc $= 2\pi\text{ cm}$

Level 4-5GCSECambridge iGCSE

## Example 2: Sector Area in terms of $\pi$

Find the area of the following sector, giving your answer in its exact form.

[3 marks]

Firstly, we need to find the overall area of the circle that the sector is part of.

The equation for the area of a circle is:

Area $= \pi r^2$

AS $r=4 \rightarrow$ Area $= \pi\times4^2 = 16\pi\text{ cm}^2$

We can now work out the area of the sector:

Area of Sector $= \dfrac{x}{360}\times$ Area of full Circle

Area of Sector $= \dfrac{36}{360}\times16\pi$

Area of Sector $= \dfrac{8}{5}\pi\text{ cm}^2$

Level 4-5GCSECambridge iGCSE

## Example 3: Arc Length

Find the length of the following arc to $3$ significant figures.

[3 marks]

Firstly, we need to find the overall circumference of the circle that the arc is part of.

The equation for the circumference of a circle is:

Circumference $= \pi d$

As the radius is $3\text{ cm}$, the diameter is $6\text{ cm}$, so the circumference is $6\pi$

We can now work out the length of the arc:

Length of Arc $= \dfrac{x}{360}\times$ Circumference of full Circle

Length of Arc $= \dfrac{100}{360}\times6\pi$

Length of Arc $= 5.24\text{ cm}$

Level 6-7GCSECambridge iGCSE

## Arc Length and Sector Area Example Questions

Firstly, find the circumference,

Circumference $= \pi d$

Circumference $= 8\pi$

And now, the length of the arc,

Length of Arc $= \dfrac{x}{360}\times$ Circumference of full Circle

Length of Arc $= \dfrac{270}{360}\times8\pi$

Length of Arc $= 6\pi\text{ cm}$

Find the area of the full circle:

$=\pi\times2^2$ $=4\pi$

Area of the sector:

$=\dfrac{230}{360}\times4\pi$ $=8.03\text{ cm}^2$

Firstly, let’s find the length of the arc.

Length of full circle circumference:

$=10\pi$

Length of arc:

$=\dfrac{150}{360}\times 10\pi$ $=\dfrac{25}{6}\pi$

Total perimeter also includes the two radii:

$\dfrac{25}{6}\pi+5+5=23\text{ cm}$ (to the nearest whole number)

Working backwards, the equation is:

$\dfrac{36}{5}\pi=\dfrac{72}{360}\times \pi r^2$

Cancel out the pi’s:

$\dfrac{36}{5}=\dfrac{72}{360}\times r^2$

Divide through by $\dfrac{72}{360}$:

$36=r^2$

$r=6$ (we can ignore the negative root as lengths are always positive)

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