GCSEMathsOCRHigherRevision Notes9. Congruence, Similarity, Vectors & TransformationsArea & Volume of Similar ShapesSimilar Areas & Volumes

Similar Areas & Volumes (OCR GCSE Maths): Revision Note

Exam code: J560

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

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Similar areas & volumes

How do I find the length, area or volume scale factors of similar shapes?

  • The scale factor (SF) for a given quantity (length, area or volume), can be found by dividing the quantity on one shape by the quantity on the other shape

    • scale space factor equals fraction numerator quantity space on space one space shape over denominator corresponding space quantity space on space the space other space shape end fraction

Two objects, A and B. Object A has a depth of 7 cm, a front surface area of 8 cm² and a volume of 56 cm³. Object B has a depth of 14 cm, a front surface area of 32 cm² and a volume of 448 cm³. Length SF = 2, area SF = 4, volume SF = 8.

  • An object could be made either bigger or smaller by a scale factor

    • When k > 1, the object is getting bigger

      • This is also true for k2 > 1 and k3 > 1

    • When 0 < k < 1, the object is getting smaller

      • This is also true for 0< k2 < 1 and 0 < k3 < 1

What is the connection between the scale factors for lengths, areas and volumes of similar shapes?

  • The length, area and volume scale factors are powers with the same base number

  • If the length scale factor is k then

    • The area scale factor is k2

    • The volume scale factor is k3

  • If you know one scale factor, you can find the scale factors

    • If you have the length scale factor

      • Area space scale space factor equals open parentheses Length space scale space factor close parentheses squared

      • Volume space scale space factor equals open parentheses Length space scale space factor close parentheses cubed

    • If you have the area scale factor

      • Length space scale space factor equals square root of Area space scale space factor end root

      • Volume space scale space factor equals open parentheses square root of Area space scale space factor end root close parentheses cubed

    • If you have the volume scale factor

      • Length space scale space factor equals cube root of Volume space scale space factor end root

      • Volume space scale space factor equals open parentheses cube root of Volume space scale space factor end root close parentheses squared

How do I find missing lengths, areas and volumes for similar shapes?

  • STEP 1
    Identify the equivalent known quantities

    • Recognise if the quantities are lengths, areas or volumes

  • STEP 2
    Find the scale factor from two known lengths, areas or volumes

    • scale space factor equals fraction numerator second space quantity over denominator first space quantity end fraction

  • STEP 3
    Use the scale factor you have found to find other required scale factor(s)

    • Length space scale space factor equals k

    • Area space scale space factor equals k squared

    • Volume space scale space factor equals k cubed

  • STEP 4
    Multiply or divide by relevant scale factor to find the missing quantity

    • Think about whether the quantity should be bigger or smaller than the given quantity

Examiner Tips and Tricks

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 write an equation.

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

Find the length scale factor k by taking the cube root of the volume scale factor k cubed

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

Substitute the value for k into formula for the heights of the similar shapes:

Height space B equals k open parentheses height space A close parentheses

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

Height of B = 15 cm

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    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.