Similar shapes is one of those topics that students often lose marks on not because they don't understand similarity — but because they forget the rules for area and volume. A simple ratio between lengths becomes squared for area and cubed for volume, and confusing these costs marks on every major exam board. AQA (8300), Edexcel (1MA1), and OCR (J560) all test this, usually in the second half of the non-calculator paper or in the geometry questions on Papers 2 and 3, often worth 3–5 marks.
What You Need to Know First
Before tackling scale factors for area and volume, make sure you're comfortable with:
- What similar shapes are — two shapes are similar if one is an enlargement of the other; all corresponding angles are equal and corresponding sides are in the same ratio
- Finding a linear scale factor — divide a length on the larger shape by the corresponding length on the smaller shape
- Basic ratio and proportion — you need to set up and solve simple equations confidently
- Volume and area formulas — for cylinders, prisms, and compound shapes, since exam questions use these shapes
- Squaring and cubing numbers — including decimals and fractions
How to Find the Scale Factor
The linear scale factor (k) is the starting point for everything. It compares a pair of corresponding lengths on two similar shapes.
k = length on larger shape ÷ corresponding length on smaller shape
Once you have k:
- Area scale factor = k²
- Volume scale factor = k³
Worked Example 1 — Area Scale Factor
Two similar triangles have base lengths of 4 cm and 10 cm.
Step 1: Find the linear scale factor.
k = 10 ÷ 4 = 2.5
Step 2: Find the area scale factor.
Area scale factor = k² = 2.5² = 6.25
Step 3: The smaller triangle has an area of 8 cm². Find the area of the larger triangle.
Area of larger triangle = 8 × 6.25 = 50 cm²
You can check this makes sense: the shape is bigger, so the area should be bigger. 50 ÷ 8 = 6.25 ✓
Worked Example 2 — Volume Scale Factor
Two similar cylinders have radii of 3 cm and 6 cm.
Step 1: Find the linear scale factor.
k = 6 ÷ 3 = 2
Step 2: Find the volume scale factor.
Volume scale factor = k³ = 2³ = 8
Step 3: The smaller cylinder has a volume of 45 cm³. Find the volume of the larger cylinder.
Volume of larger cylinder = 45 × 8 = 360 cm³
Working Backwards from Area or Volume
Exam questions often give you the areas or volumes and ask you to find a missing length. This is where students most commonly lose marks — you have to reverse the process carefully.
Working backwards from area to find k: k = √(area scale factor)
Working backwards from volume to find k: k = ∛(volume scale factor)
Worked Example 3 — Finding a Missing Length
Two similar prisms are such that the surface area of the larger is 225 cm² and the surface area of the smaller is 25 cm².
Step 1: Find the area scale factor.
Area scale factor = 225 ÷ 25 = 9
Step 2: Find the linear scale factor.
k = √9 = 3
Step 3: The smaller prism has a height of 4 cm. Find the height of the larger prism.
Height of larger prism = 4 × 3 = 12 cm
Worked Example 4 — Volume Backwards
Two similar cones have volumes of 54 cm³ and 128 cm³. The larger cone has a height of 8 cm. Find the height of the smaller cone.
Step 1: Find the volume scale factor.
Volume scale factor = 128 ÷ 54 = 64/27
Step 2: Find the linear scale factor.
k = ∛(64/27) = 4/3
Step 3: The larger cone has height 8 cm. The linear scale factor goes from smaller to larger, so:
Height of smaller cone = 8 ÷ (4/3) = 8 × (3/4) = 6 cm
Notice: k = 4/3 means the larger is 4/3 times the smaller, so the smaller is 3/4 of the larger.
Common Mistakes Students Make
Using the linear scale factor for area questions. If k = 3, the area scale factor is 9, not 3. Students see "scale factor 3" and multiply the area by 3 — wrong. Always square k for area.
Squaring instead of cubing for volume. Volume needs k³. Squaring gives the wrong answer and earns zero marks on that part. Write k² = area, k³ = volume somewhere on your paper if it helps.
Taking the square root when they should take the cube root. When working backwards from a volume ratio, you need ∛, not √. On a calculator, use the x^(1/3) button or raise to the power of (1/3).
Dividing in the wrong direction. Make sure you always divide the larger by the smaller to get k — unless the question explicitly asks you to go from larger to smaller, in which case k < 1. Keep track of which shape is which.
Forgetting units. Area is in cm² (or m²), volume is in cm³ (or m³). Losing the units or using the wrong ones can cost a mark even when the number is right.
Exam Tips for Similar Shapes
Write the formula chain at the top of your working: k → k² (area) → k³ (volume). This takes five seconds and prevents the most common error on this topic.
AQA often phrases it as "two mathematically similar..." — the word "similar" is your signal that scale factors apply. Edexcel sometimes gives the ratio of volumes and asks for a length, so practise the reverse method.
If the scale factor isn't a whole number, leave it as a fraction. Fractions are easier to cube than decimals (e.g. (2/3)³ = 8/27 is cleaner than 0.666...³).
Always check your answer is reasonable. If the shapes are similar and one is bigger, the bigger shape must have a larger area and a larger volume. If your answer says otherwise, you've divided the wrong way.
On higher tier papers, this topic is sometimes combined with algebraic expressions. For example, lengths might be given as (x + 2) and (3x − 1). Set up the scale factor equation and solve — it's just an extension of the same method.
Practise Now
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