Similar Shapes: Area and Volume Scale Factors
Similar shapes are enlargements of each other, with equal angles and corresponding lengths in the same ratio. This page covers the higher tier skill that catches most students out: while lengths scale by the linear scale factor k, areas scale by k squared and volumes scale by k cubed. Clear worked examples show how to find a missing length, work out an area or volume from a length ratio, and work backwards from two areas or two volumes to a length, followed by auto-marked practice rooms with fresh questions on every load.
Two shapes are similar when one is an enlargement of the other: the corresponding angles are equal and the corresponding lengths are all in the same ratio. That ratio is the linear scale factor, written \(k\). The important idea for the higher tier is that area and volume do not scale by \(k\): the area scales by \(k^2\) and the volume scales by \(k^3\). This page shows how to use those relationships in both directions, finding an area or volume from a length, and finding a length back from an area or volume.
The three scale factors
Find the linear scale factor first by dividing a pair of corresponding lengths (larger over smaller). Everything else follows from it.
\(k\)
The linear scale factor. Divide a length on the larger shape by the corresponding length on the smaller shape.
\(k^2\)
The area scale factor is the square of the linear scale factor. Double the lengths and the area becomes four times as big.
\(k^3\)
The volume scale factor is the cube of the linear scale factor. Double the lengths and the volume becomes eight times as big.
Why squared and cubed? Count the dimensions
The power matches the number of dimensions the measurement has. A length is one dimensional, so it scales by \(k^1=k\). An area is two lengths multiplied together (length times width), so it scales by \(k\times k=k^2\). A volume is three lengths multiplied together (length times width times height), so it scales by \(k\times k\times k=k^3\).
The picture below shows this with \(k=2\). Double a line and you get 2 copies. Double a square and each side doubles, giving \(2\times 2=4\) small squares. Double a cube and each edge doubles, giving \(2\times 2\times 2=8\) small cubes. One extra dimension means one extra factor of \(k\).
Working backwards: from an area or volume to a length
If you are given two areas or two volumes and need a length, reverse the process. The area ratio is \(k^2\), so square root it to get \(k\). The volume ratio is \(k^3\), so cube root it. Then multiply or divide a length by \(k\). For example, if two similar shapes have areas \(9\text{ cm}^2\) and \(144\text{ cm}^2\), the area ratio is \(144\div 9=16\), so \(k=4\).
\(k=\sqrt{\dfrac{\text{larger area}}{\text{smaller area}}}\)
\(k=\sqrt[3]{\dfrac{\text{larger volume}}{\text{smaller volume}}}\)
Each extra dimension brings one more factor of \(k\). Here \(k=10\), converting cm to mm:
Worked examples
💡 Example 1: a missing length
Shapes A and B are similar. Corresponding sides are 4 cm and 12 cm. A different length on A is 5 cm. Find the matching length on B.
What is happening?
Find the linear scale factor from the matching sides, then multiply the other length by it. Lengths scale by \(k\).
💡 Example 2: area from a length
Shapes A and B are similar, with sides 7 cm and 14 cm. The area of A is 15 cm squared. Find the area of B.
What is happening?
The lengths double, so the area is multiplied by \(2^2=4\), not by 2. Square the linear scale factor for areas.
💡 Example 3: volume from a length
Solids A and B are similar, with matching edges 3 cm and 6 cm. The volume of A is 40 cm cubed. Find the volume of B.
What is happening?
The lengths double, so the volume is multiplied by \(2^3=8\). Cube the linear scale factor for volumes.
💡 Example 4: working backwards from areas
Shapes A and B are similar. The area of A is 9 cm squared and the area of B is 144 cm squared. A length on A is 5 cm. Find the matching length on B.
What is happening?
The area ratio is \(k^2\). Take the square root to get the linear scale factor, then multiply the length by it.
💡 Example 5: working backwards from volumes
Solids A and B are similar. The volume of A is 25 cm cubed and the volume of B is 1600 cm cubed. An edge on A is 3 cm. Find the matching edge on B.
What is happening?
The volume ratio is \(k^3\), so take the cube root, not the square root. That single step is the only difference from Example 4. Once you have \(k\), lengths are multiplied by \(k\) as usual.
🔑 Key points
- Similar shapes have equal angles and corresponding lengths in the same ratio \(k\).
- Area scales by \(k^2\); volume scales by \(k^3\).
- Find \(k\) first from a pair of corresponding lengths.
- To go from an area back to a length, square root the area ratio. From a volume, cube root it.
- When \(k=1\) the two shapes are identical in size, not merely the same shape: that is congruence.
⚠️ Common pitfalls
- Multiplying an area by \(k\) instead of \(k^2\), or a volume by \(k\) instead of \(k^3\).
- Forgetting to square root an area ratio (or cube root a volume ratio) before using it on a length.
- Dividing the wrong way: the scale factor from small to large is larger than 1; from large to small it is less than 1.
- Assuming shapes are similar when only some sides match. Every pair of corresponding sides must share the same ratio.
- Mixing up which shape is which when reading the area or volume from the question.