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Arcs & Sectors (Cambridge O Level Additional Maths)

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Jamie W

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Jamie W

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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 equals theta over 360 space space cross times 2 pi italic space r

    • Where theta 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 straight pi, the fraction of the circle becomes fraction numerator theta over denominator 2 pi end fraction
  • Working in radians, the formula for the length of an arc will become

l equals fraction numerator theta over denominator 2 pi end fraction blank cross times 2 pi space r

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

l space equals space r theta space

    • theta 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 pi over 6 rad.

The radius of the pizza is 12 cm. Find

 

i)
the length of the outside crust of the slice of pizza (the minor arc),
 
A simple diagram will help
 
minor arc diagram for worked example
 
The formula for the length of an arc, where the angle is in radians is s equals r theta
 
s equals 12 cross times straight pi over 6
 
bold 2 bold pi bold space bold cm

 

ii)
the perimeter of the remaining pizza.
 
A diagram will help consider where the perimeter is
 
major arc diagram for worked example 
Find the angle for the major arc, by subtracting from the angle in a full circle
 
2 straight pi minus straight pi over 6 equals fraction numerator 11 straight pi over denominator 6 end fraction
 
Use the formula for the length of an arc, s equals r theta, to find the curved length of the perimeter, the major arc M
 
M equals 12 cross times fraction numerator 11 straight pi over denominator 6 end fraction equals 22 straight pi
 
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 straight pi plus 12 plus 12
 
bold 22 bold pi bold plus bold 24 bold space bold 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 equals theta over 360 cross times pi r squared

    • Where theta 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 straight pi, the fraction of the circle becomes fraction numerator theta over denominator 2 pi end fraction
  • Working in radians, the formula for the area of a sector will become

A equals fraction numerator theta over denominator 2 pi end fraction blank cross times pi space r squared

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

A equals 1 half space r squared space theta

    • theta 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 equals 1 half r squared theta

 
30 equals 1 half cross times 6 squared cross times theta

 
Solve for theta 
 

theta equals fraction numerator 30 cross times 2 over denominator 6 squared end fraction
  

bold 5 over bold 3 bold space bold radians

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    Jamie W

    Author: Jamie W

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.