Similar Areas & Volumes (Cambridge (CIE) IGCSE International Maths)

Revision Note

Written by: Amber

Reviewed by: Dan Finlay

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Two shapes are mathematically similar if one is an enlargement of the other
If the lengths of 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
The scale factor (SF) for a given quantity (length, area or volume), can be found using the formula: $scale factor = \frac{second quantity}{first quantity}$

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

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
- $scale factor = \frac{second quantity}{first quantity}$
- For two lengths, k = length SF
- For two areas, k² = area SF
- For two volumes, k³ = volume SF
STEP 3
Check the scale factor
- SF > 1 if getting bigger
- 0 < SF < 1 if getting smaller

STEP 4
If necessary, use the scale factor you have found to find other scale factors
- If you have the length scale factor
  - $Area scale factor = {(Length scale factor)}^{2}$
  - $Volume scale factor = {(Length scale factor)}^{3}$
- If you have the area scale factor
  - $Length scale factor = \sqrt{(Area scale factor)}$
  - Find the volume scale factor by finding the length scale factor first
- If you have the volume scale factor
  - $Length scale factor = \sqrt[3]{(Volume scale factor)}$
  - Find the area scale factor by finding the length scale factor first
STEP 5
Multiply or divide by relevant scale factor to find a new quantity

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

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

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

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

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:

$Height B = k (height A)$

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

Height of B = 15 cm

Last updated: 18 October 2024