O LevelMathsCambridgeRevision NotesLengths, Areas & VolumesCongruence & SimilaritySimilar Shapes

Similar Shapes (Cambridge O Level Maths)

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Similar Lengths

How do I work with similar lengths?

  • Equivalent lengths in two similar shapes will be in the same ratio and are linked by a scale factor
    • Normally the first step is to find this scale factor
    • STEP 1
      Identify equivalent known lengths
    • STEP 2
      Establish direction
      • If the scale factor is greater than 1 the shape is getting bigger 
      • If the scale factor is less than 1 the shape is getting smaller
    • STEP 3
      Find the scale factor
      • Second Length ÷ First Length
    • STEP 4
      Use scale factor to find the length you need

Examiner Tip

  • If similar shapes overlap on the diagram (or are not clear) draw them separately
    • For example, in this diagram the triangles ABC and APQ are similar:
    • Similar Triangles together, IGCSE & GCSE Maths revision notesSo we would redraw them separately before we start:Similar-Triangles-separately, IGCSE & GCSE Maths revision notes

Worked example

ABCD and PQRS are similar shapes.
Similarity – Lengths Example shapes, IGCSE & GCSE Maths revision notesFind the length of PS.

As the two shapes are mathematically similar, there will exist a value of k such that table row cell A D space end cell equals cell space k P S end cell end table and table row cell A B space end cell equals cell space k P Q end cell end table.
k is known as the scale factor.

Form an equation using the two known corresponding sides of the triangle.
 

table attributes columnalign right center left columnspacing 0px end attributes row cell A B space end cell equals cell space k P Q end cell row cell space 6 space end cell equals cell space 3 k end cell end table

Solve to find k.

table row cell k space end cell equals cell space 6 over 3 space equals space 2 end cell end table

Substitute into table row cell A D space end cell equals cell space k P S end cell end table.

table row cell A D space end cell equals cell space 2 P S end cell row cell 15 space end cell equals cell space 2 P S end cell end table

Solve to find table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell P S end cell end table.

table row cell P S space end cell equals cell space 15 over 2 end cell end table

bold italic P bold italic S bold space bold equals bold space bold 7 bold. bold 5 bold space bold cm

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Similar Areas & Volumes

What are similar shapes?

  • Two shapes are mathematically similar if one is an enlargement of the other
  • If two similar shapes are linked by the scale factor, k
    • Equivalent areas are linked by an area factor, k2
    • Equivalent volumes are linked by a volume factor, k3

How do I work with similar shapes involving area or volume?

  • STEP 1
    Identify the equivalent known quantities
    • These could be for lengths, areas or volumes
  • STEP 2
    Establish direction
    • Are they getting bigger or smaller?
  • STEP 3
    Find the Scale Factor from two known lengths, areas or volumes
    • Second Quantity ÷ First Quantity
    • Check the scale factor is > 1 if getting bigger and < 1 if getting smaller
    • If the scale factor, s.f., is from two lengths, write it as k = s.f.
    • If the scale factor, s.f., is from two areas, write it as k2 = s.f.
    • If the scale factor, s.f., is from two lengths, write it as k3 = s.f.
  • STEP 4
    Use the value of the scale factor you have found to convert other corresponding lengths, areas or volumes using
    • Area Scale Factor = (Length Scale Factor)2 
      • Or Length Scale Factor = √(Area Scale Factor)
    • Volume Scale Factor = (Length Scale Factor)3 
      • Or Length Scale Factor = ∛(Volume Length Factor) 
  • Use the scale factor to find a new quantity

Examiner Tip

  • Take extra care not to mix up which shape is which when you have started carrying out the calculations
  • It can help to label the shapes and always write an equation 
    • For example if shape A is similar to shape B:
      • length A = k(length B)
      • area A = k2(area B)
      • volume A = k3(volume B)

Worked example

Solid A and solid B are mathematically similar. 

The volume of solid A is 32 cm3.
The volume of solid B is 108 cm3.
The height of solid A is 10 cm.

Find the height of solid B.

Calculate k cubed, the scale factor of enlargement for the volumes, using volume space B space equals space k cubed open parentheses volume space A close parentheses,

Or k cubed equals fraction numerator larger space volume over denominator smaller space volume end fraction.

table row cell 108 space end cell equals cell space 32 k cubed end cell row cell k cubed space end cell equals cell space 108 over 32 space equals space 27 over 8 end cell end table
 

For similar shapes, if the volume scale factor is k cubed, then the length scale factor is k.
FInd k.

k space equals space cube root of 27 over 8 end root space equals space 3 over 2
 

Substitute into formula for the heights of the similar shapes. Height space B space equals space k open parentheses heigh t space A close parentheses,

table row cell h space end cell equals cell space 10 k end cell row cell h space end cell equals cell space 10 open parentheses 3 over 2 close parentheses space equals space 30 over 2 space equals space 15 end cell end table
 

Height of B = 15 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.