Scale Factors in Similar Figures Revision Notes | GCSE AQA Higher Maths

1. Why Scale Factors Matter

Imagine an architect who draws a building at 1:50 scale. The floor plan looks small, but the actual floor area is not 50 times bigger — it is 2500 times bigger. That gap between what you expect and what is actually true is exactly what this lesson is about.

When two shapes are similar, their lengths scale by one factor, their areas scale by a different factor, and their volumes scale by yet another. Confusing these three is one of the most reliable ways to drop marks on an Higher-tier paper. By the end of this lesson, you will know exactly which factor to reach for and why.

2. The Linear Scale Factor

Finding $k$ is always your first move. Pick any pair of corresponding sides, divide the new measurement by the original, and that ratio is your linear scale factor. Every other calculation in this topic flows from that single number.

In the diagram above, Shape A has width 3 cm and Shape B has width 6 cm. Dividing: $k = \frac{6}{3} = 2$. You can verify this with the heights: $\frac{4}{2} = 2$. Both pairs of corresponding lengths give the same ratio — that consistency is the hallmark of similar figures.

3. From Lengths to Areas — The Area Scale Factor

Here is where most students trip up. If every length doubles, you might guess the area doubles too. Work through the rectangle above and you will see why that is wrong.

Shape A has area $3 \times 2 = 6 \text{ cm}^2$. Shape B has area $6 \times 4 = 24 \text{ cm}^2$. The area has multiplied by $\frac{24}{6} = 4$, which is $2^2$. The linear scale factor was 2, so the area scale factor is $2^2 = 4$. This is not a coincidence — it is always true.

Think of it this way: area is two-dimensional, so the scale factor gets applied in two directions simultaneously — once for width and once for height. That is why you square it. The same logic extends to volume, which is three-dimensional, meaning we apply the scale factor in three directions: length, width, and height.

4. From Lengths to Volumes — The Volume Scale Factor

Volume is three-dimensional, so the scale factor gets applied in three directions. If the linear scale factor is $k$, the volume scale factor is $k^3$. A shape with $k = 3$ has a volume $3^3 = 27$ times larger — not 3 times, not 9 times, but 27 times.

A useful anchor: picture a cube with side length 1 cm scaled up by factor $k$. Its new side is $k$ cm. Its new volume is $k \times k \times k = k^3$ cm$^3$. The cube makes the reasoning concrete, but the rule applies to any similar solid — spheres, cones, pyramids, or composite shapes.

5. Worked Examples

Theory is clear, but exam questions wrap these ideas in context. The following walkthroughs show the full method — including how marks are awarded — so you can see exactly what the examiner expects on paper. This journey illustrates how linear, area, and volume scale factors are calculated in practice.

Worked example: Finding an Unknown Area in Similar Shapes

Two similar triangles, P and Q, have corresponding base lengths of 4 cm and 10 cm. Triangle P has area 12 cm². Calculate the area of triangle Q.

1. $k = \dfrac{10}{4} = 2.5$ — Divide the new (larger) length by the original length. [M1]
2. $k^2 = 2.5^2 = 2.5 \times 2.5 = 6.25$ — Square the linear scale factor. Expand longhand: $2.5 \times 2.5 = 6.25$. [M1]
3. $\text{Area of Q} = 12 \times 6.25 = 75 \text{ cm}^2$ — Apply the area scale factor to the known area. [A1]
4. $\text{Area of Q} = 75 \text{ cm}^2$ — Always include units — missing units can cost a mark on GCSE Higher exam papers.

Answer: 75 cm²

Worked example: Finding an Unknown Volume in Similar Solids

Two similar cylinders, A and B, have heights 3 cm and 9 cm respectively. Cylinder A has volume 40 cm³. Calculate the volume of cylinder B.

1. $k = \dfrac{9}{3} = 3$ — Heights are corresponding lengths, so divide new by original. [M1]
2. $k^3 = 3^3 = 3 \times 3 \times 3 = 27$ — Cube the linear scale factor. Expanding: $3 \times 3 = 9$, then $9 \times 3 = 27$. [M1]
3. $\text{Volume of B} = 40 \times 27 = 1080 \text{ cm}^3$ — Apply the volume scale factor to the known volume. [A1]
4. $\text{Volume of B} = 1080 \text{ cm}^3$ — Cubic units are required here — cm³, not cm² or cm.

Answer: 1080 cm³

Worked example: Working Backwards — Finding a Length from Areas

Two similar shapes have areas of 50 cm² and 200 cm². The smaller shape has a perimeter of 30 cm. Find the perimeter of the larger shape.

1. $\text{Area SF} = \dfrac{200}{50} = 4$ — Divide the larger area by the smaller area. [M1]
2. $k = \sqrt{4} = 2$ — Take the square root of the area scale factor to get back to lengths. [M1]
3. $\text{Perimeter of larger shape} = 30 \times 2 = 60 \text{ cm}$ — Perimeter is a length, so multiply by $k$ (not $k^2$). [A1]
4. $\text{Perimeter} = 60 \text{ cm}$ — Perimeter is measured in cm (linear units), not cm².

Answer: 60 cm

6. Deliberate Practice

Now it is your turn. Work through each problem before checking the solution. Pay attention to whether the question asks for a length, an area, or a volume — that tells you which scale factor to use.

7. Exam Technique and Common Mistakes

The three scale factors — $k$, $k^2$, and $k^3$ — are straightforward once you have the right one in hand. The difficulty on GCSE Higher exam papers is that questions are designed to make you reach for the wrong one. Here is how to stay sharp.

One more pattern worth knowing: Exam questions sometimes gives you two volumes and asks for a length. The route is always the same — find the volume scale factor, cube-root it to get $k$, then apply $k$ to the known length. Never skip the step of finding $k$ explicitly.

This final question brings together all three scale factors in one problem — exactly the kind of multi-step question that appears at the top end of Higher-tier papers. Work through it carefully before checking the marking points.