Similar Areas & Volumes (OCR GCSE Maths)

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Naomi C

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Naomi C

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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 = 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 and solid are mathematically similar. 

The volume of solid is 32 cm3.
The volume of solid B is 108 cm3.
The height of solid 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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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.