Similar Areas & Volumes (Edexcel IGCSE Maths)

Spec point — 4.11C

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) between two similar shapes can be found by dividing the quantity on one shape by the quantity on the other shape

  • $\mathrm{scale} \mathrm{factor}=\frac{\mathrm{quantity} \mathrm{on} \mathrm{one} \mathrm{shape}}{\mathrm{corresponding} \mathrm{quantity} \mathrm{on} \mathrm{the} \mathrm{other} \mathrm{shape}}$

Example of scale factor of similar shapes

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

  • The object gets bigger if the scale factor is bigger than 1
  • The object gets smaller if the scale factor is bigger than 0 and less than 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

    • $Area \text{scale} \text{factor} = ( Length \text{scale} \text{factor} ) ^{2}$
    • $Volume \text{scale} \text{factor} = ( Length \text{scale} \text{factor} ) ^{3}$

  • If you have the area scale factor

    • $\mathrm{Length} \mathrm{scale} \mathrm{factor}=\sqrt{\mathrm{Area} \mathrm{scale} \mathrm{factor}}$
    • $\text{Volume} \text{scale} \text{factor} = ( \sqrt{Area \text{scale} factor} )^{3}$

  • If you have the volume scale factor

    • $\text{Length} \text{scale} \text{factor} = \sqrt[3]{Volume \text{scale} factor}$
    • $\text{Area} \text{scale} \text{factor} = ( \sqrt[3]{Volume \text{scale} factor} )^{2}$

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

  • $\mathrm{scale} \mathrm{factor}=\frac{\mathrm{second} \mathrm{quantity}}{\mathrm{first} \mathrm{quantity}}$

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

  • $\mathrm{Length} \mathrm{scale} \mathrm{factor}=k$
  • $\mathrm{Area} \mathrm{scale} \mathrm{factor}=k^{2}$
  • $\mathrm{Volume} \mathrm{scale} \mathrm{factor}=k^{3}$

• 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

Exam Hint 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.

Answer:

Calculate $k^{3}$, the scale factor of enlargement for the volumes, using: $\text{volume} B = k ^{3} ( \text{volume} A )$

Or $k^{3}=\frac{\mathrm{larger} \mathrm{volume}}{\mathrm{smaller} \mathrm{volume}}$

$\begin{gathered} 108 = 32 k^{3} \\ k^{3} & = \frac{108}{32} = \frac{27}{8} \end{gathered}$

Find the length scale factor $k$ by taking the cube root of the volume scale factor $k^{3}$

$k = \sqrt[3]{\frac{27}{8}} = \frac{3}{2}$

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

$Height B = k ( \text{height} A )$

$\begin{gathered} h = 10 k \\ h = 10 ( \frac{3}{2} ) = \frac{30}{2} = 15 \end{gathered}$

Final answer: Height of B = 15 cm