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Arcs & Sectors (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Length of an Arc

What is an arc?

  • An arc is a part of the circumference of a circle

    • You can think of it as the crust on a slice of pizza

  • The length of an arc depends on

    • the size of the angle at the centre of the circle

    • the radius of the circle

  • If the angle at the centre is less than 180° then the arc is known as a minor arc

  • If the angle at the centre is more than 180° then the arc is known as a major arc

How do I find the length of an arc using degrees?

  • The length of an arc is simply a fraction of the total 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 equals theta over 360 cross times 2 pi italic space r

  • theta is the angle measured in degrees

  • r is the radius

  • This is not on the exam formula sheet, so you need to remember it

How do I find the length of an arc using radians?

  • When working in radians the formula for arc length is much simpler

    • This is because pi is already 'built into' radians

  • The formula for the length, l, of an arc is

    • l equals r theta

  • theta is the angle measured in radians

  • r is the radius

  • This is not on the exam formula sheet, so you need to remember it

Examiner Tips and Tricks

  • Be careful on arc length questions

    • Finding the arc length may only be part of a larger question

    • For example finding the total perimeter of a circle sector

  • Make sure you are using the formula that matches the angle measure (degrees or radians)

Worked Example

A circular pizza has had a slice cut from it. The slice is in the shape of a sector, with the angle at the centre being 38°. The radius of the pizza is 12 cm.

(a) Find the length of the outside crust of the slice of pizza (the minor arc), giving your answer correct to 3 significant figures.

Drawing a diagram can help with questions like this

Minor sector with minor arc, angle and radius


Use the arc length formula (in the degrees form) with r equals 12 and theta equals 38
(You could also convert 38 degree to radians and use the radians version of the formula)

l equals 38 over 360 cross times 2 pi open parentheses 12 close parentheses equals fraction numerator 38 pi over denominator 15 end fraction equals 7.958701...

Round to 3 significant figures

(Note that fraction numerator 38 pi over denominator 15 end fraction cm is the exact value answer)

bold 7 bold. bold 96 bold space bold cm bold space stretchy left parenthesis 3 space s. f. stretchy right parenthesis 

(b) Find the perimeter of the remaining pizza, giving your answer correct to 3 significant figures.

Drawing a diagram can help

The remaining pizza will be in the shape of a major sector

The angle at the centre will be  360 minus 38 equals 322 degree

Major sector of circle with radius and angle at centre

The perimeter will include both the major arc and the two radii

Use the arc length formula (in the degrees form) with r equals 12 and theta equals 322

perimeter equals open parentheses 322 over 360 cross times 2 pi open parentheses 12 close parentheses close parentheses plus 12 plus 12 equals fraction numerator 322 pi over denominator 15 end fraction plus 24 equals 91.439522...

Round to 3 significant figures

(Note that fraction numerator 322 pi over denominator 15 end fraction plus 24 cm is the exact value answer)

91.4 cm (3 s.f.)

Worked Example

A sector of a circle has a radius of 7cm and an angle at the centre of straight pi over 6 radians. Find the perimeter of the sector, giving your answer as an exact value.

Drawing a diagram can help

Circle sector with radius and angle at centre


The perimeter of sector will include both the length of the arc and the two radii

Use the arc length formula (in the radians form) with r equals 7 and theta equals pi over 6

perimeter equals open parentheses 7 close parentheses open parentheses pi over 6 close parentheses plus 7 plus 7 equals fraction numerator 7 pi over denominator 6 end fraction plus 14

The question asks for an exact value answer, so just write that down (with units!)

fraction numerator bold 7 bold italic pi over denominator bold 6 end fraction bold plus bold 14 bold space bold space bold cm

Area of a Sector

What is a sector?

  • A sector is a part of a circle enclosed by two radii (radiuses) and an arc

    • You can think of this as the shape of a single slice of pizza

  • The area of a sector depends on

    • the size of the angle at the centre of the sector

    • the radius of the circle

  • If the angle at the centre is less than 180° then the sector is known as a minor sector

  • If the angle at the centre is more than 180° then the sector is known as a major sector

How do I find the area of a sector using degrees?

  • 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 equals theta over 360 cross times pi r squared

  • theta is the angle measured in degrees

  • r is the radius

  • This is not on the formula sheet, so you need to remember it

How do I find the area of a sector using radians?

  • When working in radians the formula for sector area is much simpler

    • This is because pi is already 'built into' radians

  • The formula for the area, A, of a sector is

    • A equals 1 half r squared theta

  • theta is the angle measured in radians

  • r is the radius

  • This is not on the formula sheet, so you need to remember it

Examiner Tips and Tricks

  • Make sure you are using the formula that matches the angle measure (degrees or radians)

Worked Example

Jamie has divided a circle of radius 50 cm into two sectors: a minor sector of angle 100°, and a major sector of angle 260°. He is going to paint the minor sector blue and the major sector yellow.

(a) Find the area Jamie will paint blue, giving your answer correct to 3 significant figures

Drawing a diagram can be helpful in questions like this

Circle with minor and major sectors


Use the sector area formula (degrees form) with r equals 50 and theta equals 100

(You could also convert 100 degree to radians and use the radians version of the formula)

A equals 100 over 360 cross times pi open parentheses 50 close parentheses squared equals fraction numerator 6250 pi over denominator 9 end fraction equals 2181.6615...


Round to 3 significant figures

(Note that fraction numerator 6250 pi over denominator 9 end fraction cm2 is the exact value answer) 

bold 2180 bold space bold cm to the power of bold 2 bold space bold space stretchy left parenthesis 3 space s. f. stretchy right parenthesis

(b) Find the area Jamie will paint yellow, giving your answer correct to 3 significant figures.

This will be the rest of the circle from the preceding diagram

Use the sector area formula (degrees form) with r equals 50 and theta equals 260

A equals 260 over 160 cross times pi open parentheses 50 close parentheses squared equals fraction numerator 16250 pi over denominator 9 end fraction equals 5672.3200...

Round to 3 significant figures

(Note that fraction numerator 16250 pi over denominator 9 end fraction cm2 is the exact value answer)

bold 5670 bold space bold cm to the power of bold 2 bold space bold space begin bold style stretchy left parenthesis 3 space s. f. stretchy right parenthesis end style

Worked Example

A sector of a circle has a radius of 7 cm and an angle at the centre of straight pi over 6 radians. Find the area of the sector, giving your answer as an exact value.

Drawing a diagram can help

Circle sector with radius and angle a centre


Use the sector area formula (radians form) with r equals 7 and theta equals pi over 6

A equals 1 half open parentheses 7 close parentheses squared open parentheses pi over 6 close parentheses equals fraction numerator 49 pi over denominator 12 end fraction


The question asks for an exact value answer, so just write that down (with units!)

fraction numerator bold 49 bold pi over denominator bold 12 end fraction bold space bold cm to the power of bold 2

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.