Similar Areas & Volumes

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

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

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)

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

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

For similar shapes, if the volume scale factor is $k^{3}$ , then the length scale factor is $k$ .
FInd $k$ .

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

Substitute into formula for the heights of the similar shapes. $Height B = k (heigh t A)$ ,

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

Height of B = 15 cm

Similar Areas & Volumes (OCR GCSE Maths)