Arcs & Sectors (Edexcel GCSE Maths: Foundation)

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Jamie W

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Jamie W

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Arc Lengths & Sector Areas

What is an arc?

  • An arc is a part of the circumference of a circle 
  • Two points on a circumference of a circle will create two arcs 
    • The smaller arc is known as the minor arc
    • The bigger arc is known as the major arc

How do I find the length of an arc?

  • The angle formed in a sector by the two radii is often labelled θ (the Greek letter “theta”)
  • You can calculate the length of an arc by adapting the formulae for the circumference of a circle
    • A full circle is equal to 360° so the portion of the circle will be the angle, θ°, out of 360°
      • Length space of space an space arc space equals space theta over 360 cross times 2 pi italic space r
      • You need to remember this formulae; it is not given in the exam
  • STEP 1
    Divide the angle by 360 to form a fraction
    • theta over 360
  • STEP 2
    Calculate the circumference of the full circle
    • 2 straight pi r
  • STEP 3
    Multiply the fraction by the circumference
    • theta over 360 cross times 2 straight pi r

What is a sector?

  • A sector is an area of a circle enclosed by two radii (radiuses) and an arc
    • A sector looks like a slice of a circular pizza
    • The curved edge of a sector is the arc
  • Two radii in a circle will create two sectors
    • The smaller sector is known as the minor sector
    • The bigger sector is known as the major sector

How do I find the area of a sector?

  • You can calculate the area of a sector by adapting the formulae for the area of a circle
    • A full circle is equal to 360° so the portion of the circle will be the angle, θ°, out of 360°
      • Area space of space straight a space Sector space equals space theta over 360 cross times pi italic space r squared
      • You need to remember this formulae; it is not given in the exam
  • STEP 1
    Divide the angle by 360 to form a fraction
    • theta over 360
  • STEP 2
    Calculate the area of the full circle
    • straight pi r squared
  • STEP 3
    Multiply the fraction by the area
    • theta over 360 cross times straight pi r squared

Sector Area & Arc Length Formulae

Examiner Tip

  • To help remember the formulae, just remember they are a fraction of a circle's area, or a fraction of a circle's circumference

Worked example

A sector of a circle is shown.

A sector

The angle, θ, is 72° and the radius, r, is 5 cm.

(a)
Find the area of the sector, giving your answer correct to 3 significant figures.
 
Substitute θ  = 72° and = 5 into the formula for the area of a sector, A equals theta over 360 pi space r squared 
 
A equals space 72 over 360 pi cross times 5 squared space
 
Use a calculator to work out this value
 
15.70796...
 
Round your answer to 3 significant figures
15.7 cm2
   
(b)
Find the length of the arc of the sector, giving your answer as a multiple of pi.
 
Substitute θ  = 72° and = 5 into the formula for the length of an arc, l space equals space theta over 360 2 pi space r 
 
l space equals space 72 over 360 cross times 2 cross times straight pi cross times 5
 
Simplify the number part without pi
 
72 over 360 cross times 2 cross times 5 equals 1 fifth cross times 10 equals 2
 
Write down the final answer with pi
bold 2 bold italic pi cm

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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.