One shape, three scale factors
Imagine a small box and a bigger box that is exactly the same shape, just scaled up. Every length on the big box is, say, twice the matching length on the small one. It is tempting to assume the area and the volume also just double. They do not.
Area is a length multiplied by a length, so it grows by the scale factor twice over. Volume is a length multiplied by a length multiplied by a length, so it grows by the scale factor three times over. The plain idea: stretch a shape and its skin grows faster than its edges, and its contents grow faster still.
- Length scale factor
- The number \(k\) that every length is multiplied by to go from the smaller similar shape to the larger one.
The three rules
If the length scale factor is \(k\), then the area and volume scale factors follow from it directly:
| Quantity | Multiply by | Example, \(k=2\) |
|---|---|---|
| Length | \(k\) | \(\times 2\) |
| Area | \(k^2\) | \(\times 4\) |
| Volume | \(k^3\) | \(\times 8\) |
Count the dimensions
If you ever forget the powers, count how many lengths make the quantity. Area is 2 lengths, so \(k^2\). Volume is 3 lengths, so \(k^3\). The power matches the number of dimensions.
A worked example
A small model has surface area \(8\ \text{cm}^2\) and volume \(3\ \text{cm}^3\). A larger similar model is built with a length scale factor of \(4\). Find the surface area and the volume of the larger model.
Working backwards
Exam questions often hand you the area or volume ratio and ask for the length scale factor. Just reverse the power. If you know the area scale factor, take its square root. If you know the volume scale factor, take its cube root.
Where this is assessed
Choosing \(k^2\) or \(k^3\) correctly is Criterion A (Knowing and understanding). Laying the reasoning out so a reader can follow which factor you used and why is Criterion C (Communicating). Applying it to real solids, scale models, packaging or maps, sits under Criterion D (Applying maths in context).
Check yourself
1. Two similar cones have a length scale factor of 3. What is the volume scale factor? +
Volume uses the cube of the length scale factor: \(3^3 = \mathbf{27}\). The larger cone holds 27 times as much.
2. A small shape has area 10 cm² and a similar larger shape has area 90 cm². Find the length scale factor. +
Area scale factor \(= \dfrac{90}{10} = 9\). The length scale factor is the square root: \(\sqrt{9} = \mathbf{3}\).
3. A length scale factor of 2 links two similar solids. The smaller has volume 5 cm³. Find the volume of the larger. +
Volume scale factor \(= 2^3 = 8\). So the larger volume \(= 5 \times 8 = \mathbf{40}\ \text{cm}^3\).
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.