Similar Shapes: Area and Volume Scale Factors
One idea, three versions. If lengths scale by k, then areas scale by k² and volumes by k³. Almost every mark lost in this topic comes from using the length scale factor where the squared or cubed one was needed.
1. The rule
Length scale factor: k Area scale factor: k² Volume scale factor: k³
Why: area is length × length, so it picks up the factor twice. Volume is length × length × length, so three times. If a shape is twice as long, it has 4× the area and 8× the volume.
| If lengths ×2 | If lengths ×3 | If lengths ×½ |
|---|---|---|
| areas ×4 | areas ×9 | areas ×¼ |
| volumes ×8 | volumes ×27 | volumes ×⅛ |
Doubling the lengths does not double the area. This is the single most common misconception here, and understanding why is more reliable than memorising the table.
2. Finding the scale factor
k = new length ÷ corresponding original length
Corresponding matters — match the side that plays the same role in each shape.
Check with a second pair of sides. If two pairs give different values, either the shapes aren’t similar or you’ve matched the wrong sides.
Converting between the three scale factors:
| Given | To get k | |
|---|---|---|
| Length ratio | k directly | |
| Area ratio | √ of it | k = √(area SF) |
| Volume ratio | ∛ of it | k = ∛(volume SF) |
Square root for areas, cube root for volumes. Going from lengths you square or cube; coming back you take the root. Getting these the wrong way round was recorded repeatedly.
3. Working forwards
Example: two similar triangles, with sides 4 cm and 10 cm corresponding. The smaller has area 12 cm². Find the area of the larger.
- k = 10 ÷ 4 = 2.5
- Area scale factor = k² = 6.25
- Larger area = 12 × 6.25 = 75 cm²
Example: two similar cones, heights 3 cm and 6 cm. The smaller has volume 20 cm³.
- k = 6 ÷ 3 = 2
- Volume scale factor = k³ = 8
- Larger volume = 20 × 8 = 160 cm³
Multiply by k² for area, k³ for volume — never by k. Using the linear scale factor for an area was the most frequent error in lessons.
4. Working backwards
This is the harder direction, and the one worth practising.
Example: two similar solids have volumes 27 cm³ and 64 cm³. The smaller has height 6 cm. Find the height of the larger.
- Volume scale factor = 64/27
- k = ∛(64/27) = 4/3
- Larger height = 6 × 4/3 = 8 cm
Example: two similar shapes have areas 20 cm² and 45 cm². The smaller has a side of 4 cm.
- Area scale factor = 45/20 = 2.25
- k = √2.25 = 1.5
- Larger side = 4 × 1.5 = 6 cm
Take the root FIRST, then multiply the length. Multiplying the length by the area or volume ratio directly is the classic error — it gives an answer far too large.
5. Method
- Identify whether you’re dealing with lengths, areas or volumes — check the units: cm, cm², cm³
- Find k, taking a root if you were given an area or volume ratio
- Apply k, k² or k³ as appropriate
- Check the answer is sensible — the larger shape must have the larger value
The units tell you which scale factor you need. cm² means area, so k². cm³ means volume, so k³. This one check prevents most errors in the topic.
Sketch both shapes and label them. Tutors recommended this consistently — it makes corresponding sides obvious.
Convert to the same units first. Comparing cm with m gives a wrong scale factor.
6. When are shapes similar?
Two shapes are similar if one is an enlargement of the other: corresponding angles are equal and corresponding sides are in the same ratio.
Congruent shapes are identical — a special case where k = 1.
An enlargement always produces a similar shape; reflections, rotations and translations produce congruent ones.
7. Mistakes that cost marks
Using k instead of k² for area, or instead of k³ for volume.
Multiplying a length by an area or volume ratio.
Forgetting to take the square or cube root when working backwards.
Taking a square root where a cube root was needed.
Matching non-corresponding sides.
Not converting units before finding k.
Ignoring the units and so choosing the wrong scale factor.
Inverting the scale factor, making the larger shape smaller.
Frequently asked questions
What is the area scale factor? k², where k is the length scale factor.
What is the volume scale factor? k³.
If lengths double, what happens to the area? It becomes 4 times larger.
If lengths double, what happens to the volume? It becomes 8 times larger.
How do I find k from an area ratio? Take the square root.
How do I find k from a volume ratio? Take the cube root.
How do I find a length given two volumes? Cube-root the volume ratio to get k, then multiply the known length by it.
When are two shapes similar? When corresponding angles are equal and corresponding sides are in the same ratio.
What’s the difference between similar and congruent? Congruent shapes are identical (k = 1); similar shapes are the same shape, different size.
Quick revision checklist
- I know lengths scale by k, areas by k², volumes by k³
- I understand why — area is two lengths, volume three
- I can find k from corresponding sides
- I check k with a second pair of sides
- I take a square root to get k from an area ratio
- I take a cube root to get k from a volume ratio
- I take the root before multiplying a length
- I use the units to decide which scale factor I need
- I convert units before finding k
- I check the larger shape has the larger value
- I know the difference between similar and congruent
These notes cover similar shapes and their area and volume scale factors in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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