Arcs & Sectors (Cambridge O Level Additional Maths)

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Jamie W

Last updated

20 September 2024

Length of an Arc

What is an arc?

An arc is a part of the circumference of a circle
- It is easiest to think of it as the crust of a single slice of pizza
The length of an arc depends of the size of the angle at the centre of the circle
If the angle at the centre is less than 180° then the arc is known as a minor arc
- This could be considered as the crust of a single slice of pizza
If the angle at the centre is more than 180° then the arc is known as a major arc
- This could be considered as the crust of the remaining pizza after a slice has been taken away

How do I find the length of an arc?

The length of an arc is simply a fraction of the circumference of a circle
- The fraction can be found by dividing the angle at the centre by 360°
The formula for the length, $l$ , of an arc is

$l = \frac{θ}{360} \times 2 π r$

- Where $θ$ is the angle measured in degrees
- $r$ is the radius

How do I use radians to find the length of an arc?

As the radian measure for a full turn is $2 π$ , the fraction of the circle becomes $\frac{θ}{2 π}$
Working in radians, the formula for the length of an arc will become

$l = \frac{θ}{2 π} \times 2 π r$

Simplifying, the formula for the length, $l$ , of an arc is

$l = r θ$

- $θ$ is the angle measured in radians
- $r$ is the radius

Worked example

A circular pizza has had a slice cut from it, the angle of the slice that was cut was $\frac{π}{6}$ rad.

The radius of the pizza is 12 cm. Find

the length of the outside crust of the slice of pizza (the minor arc),

A simple diagram will help

The formula for the length of an arc, where the angle is in radians is

s = r θ

s = 12 \times \frac{π}{6}

2 π cm

ii)

the perimeter of the remaining pizza.

A diagram will help consider where the perimeter is

Find the angle for the major arc, by subtracting from the angle in a full circle

2 π - \frac{π}{6} = \frac{11 π}{6}

Use the formula for the length of an arc,

s = r θ

, to find the curved length of the perimeter, the major arc

M

M = 12 \times \frac{11 π}{6} = 22 π

As we are finding the perimeter of the whole shape, we need to add on the two straight lengths formed by the slice which has been cut out

22 π + 12 + 12

22 π + 24 cm

Unless asked to otherwise, it is best to give answers in an exact form

Area of a Sector

What is a sector?

A sector is a part of a circle enclosed by two radii (radiuses) and an arc
- It is easier to think of this as the shape of a single slice of pizza
The area of a sector depends of the size of the angle at the centre of the sector
If the angle at the centre is less than 180° then the sector is known as a minor sector
- This could be considered as the shape of a single slice of pizza
If the angle at the centre is more than 180° then the sector is known as a major sector
- This could be considered as the shape of the remaining pizza after a slice has been taken away

How do I find the area of a sector?

The area of a sector is simply a fraction of the area of the whole circle
- The fraction can be found by dividing the angle at the centre by 360°
The formula for the area, $A$ , of a sector is

$A = \frac{θ}{360} \times π r^{2}$

- Where $θ$ is the angle measured in degrees
- $r$ is the radius

How do I use radians to find the area of a sector?

As the radian measure for a full turn (360°) is $2 π$ , the fraction of the circle becomes $\frac{θ}{2 π}$
Working in radians, the formula for the area of a sector will become

$A = \frac{θ}{2 π} \times π r^{2}$

Simplifying, the formula for the area, $A$ , of a sector is

$A = \frac{1}{2} r^{2} θ$

- $θ$ is the angle measured in radians
- $r$ is the radius

Examiner Tip

These formulae are not given to you - you need to remember them!
Make sure that you read the question carefully to determine exactly what you need to calculate
- the arc length or area of a sector,
- the perimeter or area of a compound shape,
- or something else that incorporates the arc length or area!

Worked example

A sector of radius 6 cm has an area of 30 cm².
Find the angle at the centre of the sector in radians.

Draw a diagram to help

sector area diagram for worked example

Use the formula for the area of a sector, where the angle is in radians; $A = \frac{1}{2} r^{2} θ$

$30 = \frac{1}{2} \times 6^{2} \times θ$

Solve for $θ$

$θ = \frac{30 \times 2}{6^{2}}$

$\frac{5}{3} radians$

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