Syllabus Edition

First teaching 2023

First exams 2025

|

Arcs & Sectors (CIE IGCSE Maths: Core)

Revision Note

Test yourself

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

What is a sector?

  • A sector is the part 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

What formulae do I need to know?

  • You need to be able to calculate the length of an arc and the area of a sector
  • The angle formed in a sector by the two radii is often labelled θ (the Greek letter “theta”)
  • You can calculate the area of a sector or the length of an arc by adapting the formulae for the area or circumference of a circle
    • A full circle is equal to 360° so the fraction will be the angle, θ°, out of 360°
      • Area space of space straight a space sector equals theta over 360 cross times pi italic space r squared
      • Arc space length space equals space theta over 360 cross times 2 pi italic space r

Sector Area & Arc Length Formulae

  • Working with sector and arc formulae is just like working with any other formula:
    • Write down what you know (or what you want to know)
    • Pick the correct formula
    • Substitute the values in and solve

How do I find the length of an arc?

  • 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

How do I find the area of a sector?

  • 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

Examiner Tip

  • The area and circumference of a circle formulae are given to you in the exam.
    • You just need to remember how to find the correct fraction of the whole circle.

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

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.