Arcs & Sectors (Edexcel IGCSE Maths)

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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

What is a sector?

  • In technical terms, a sector is the part of a circle enclosed by two radii (radiuses) and an arc
  • It’s much easier to think of a sector as the shape of a slice of a circular pizza (or cake, or pie, or …) and an arc as the curvy bit at the end of it (where the crust is)
  • 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

Arc length and area of a sector formula

  • If the angle of the slice is θ (the Greek letter “theta”) then the formulae for the area of a sector and the length of an arc are just fractions of the area and circumference of a circle:
    • Remember that a full circle is equal to 360° so the fraction will be the angle, out of 360

Sector Area & Arc Length Formulae, IGCSE & GCSE Maths revision notes

  • If you are not too good at remembering formulae there is a logic to these two
    • You’ll need to remember the circumference and area formulas
    • After that we are just finding a fraction of the whole circle – “θ out of 360”
  • Working with sector and arc formulae is just like working with any other formula
    • WRITE DOWN – what you know (what you want to know)
    • Pick correct FORMULA
    • SUBSTITUTE and SOLVE

Examiner Tip

  • If you’re under pressure and can’t remember which formula is which, remember that area is always measured in square units (cm2, m2 etc.) so the formula with r2 in it is the one for area
  • The length of an arc is just a length, so its units will be the same as for length (cm, m, etc)

Worked example

AOB  is a sector of a circle with angle 42°, as shown. 

Sector-42deg, IGCSE & GCSE Maths revision notes

The area of the sector AOB is 28 cm2.

(a)
Find the radius of the circle, giving your answer correct to 2 decimal places.
 
We know the area and the angle and want to find the radius so we will need to substitute the information into the area of a sector formula and solve to find the radius.
Substitute A = 28 and θ = 42° into the area of a sector formula, A space equals space theta over 360 pi r squared .
28 space equals space 42 over 360 straight pi open parentheses r close parentheses squared space
Simplify.
 28 space equals space 7 over 60 straight pi r squared space
Rearrange to find r.
table row cell fraction numerator 28 space cross times space 60 over denominator 7 end fraction space end cell equals cell space pi r squared space end cell row cell pi r squared space end cell equals 240 row cell r squared space end cell equals cell 240 over pi end cell row cell r space end cell equals cell space square root of 240 over pi end root end cell row blank equals cell space 8.74038... end cell end table
Round to 2 decimal places.
The second decimal place has the value 4 and the third decimal place is a 0, which is less than 5 so do not change the value of the second decimal place.
bold italic r bold space bold equals bold space bold 8 bold. bold 74 bold space bold cm
 
(b)
Find the length of the arc AB, giving your answer correct to the nearest whole number.
 
We know the radius and the angle so we can substitute the information into the formula for the length of an arc.
Substitute r = 8.7403... and θ = 42° into the formula for the length of an arc, l space equals space theta over 360 2 pi r .
l space equals space 42 over 360 open parentheses 2 straight pi open parentheses 8.7403... close parentheses close parentheses
Use your calculator to find the answer, type the radius in carefully or use the memory function on your calculator.
 table row cell l space end cell equals cell space 7 over 60 open parentheses 17.48... straight pi close parentheses end cell row blank equals cell space 6.4070... space end cell end table
Round to the nearest whole number.
The units place has the value 6 and the first decimal place is a 4, which is less than 5 so do not change the value of the units.
bold italic l bold space bold equals bold 6 bold space bold cm

 

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.