If two similar solids have a volume ratio of 8 to 1, what is the ratio of their lengths?

2 to 1. The volume ratio is the length ratio cubed, so to recover the length ratio you take the cube root: the cube root of 8 is 2. You do not divide by 3, and you do not reuse the volume number. A volume ratio of 27 to 1 gives a length ratio of 3 to 1.

A prism has volume 72 cm cubed. A similar prism has volume 1125 cm cubed. What is the linear scale factor?

2.5. The volume scale factor is 1125 divided by 72 = 15.625, and the linear scale factor is the cube root of that, because volume is the linear factor cubed. The cube root of 15.625 is 2.5. The trap is to scale a length by 15.625 itself, skipping the cube root.

Two similar shapes have areas in the ratio 9 to 16. What is the ratio of their lengths?

3 to 4. Area is the length factor squared, so the length ratio is the square root of the area ratio: the square root of 9 to the square root of 16 is 3 to 4. Leaving the answer as 9 to 16 forgets the square root.

GCSE Maths Higher · AQA · Similar shapes & scaling

Reverse and cross-dimension scaling

Forward scaling is well drilled: a length factor k gives an area factor $k^2$ and a volume factor $k^3$ . The Higher discriminator is going backward. To recover a length from a volume you take a cube root, not a division; to recover a length from an area you take a square root. A volume ratio of 8 to 1 is a length ratio of 2 to 1, because $\sqrt[3]{8} = 2$ .

The second half of the trap is cross-dimension translation. The length, area and volume ratios are three different powers of the same factor, so you can never reuse one as another. A volume ratio of 1 to 2 does not make the surface area ratio 1 to 2, and if the two solids are not even mathematically similar there is no single scale factor at all: every ratio has to be computed directly.

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How to spot it in your own work

You divided a volume ratio by 3 (or reused it unchanged) to get a length ratio, instead of taking the $\sqrt[3]{\,}$ cube root.
You scaled a length by a volume scale factor, multiplying by 15.625 instead of its cube root 2.5.
You copied a volume ratio onto a surface area, when the two are different dimensions and must be worked out separately.
You handed back an area ratio as the length ratio, giving 9 to 16 instead of $\sqrt{9} : \sqrt{16} = 3 : 4$ .

An exam question that triggers it

Here is the structure of JUN24 Paper 2 Question 25:

A triangular prism has a right-angled cross-section with legs 3 cm and 4 cm and length 12 cm. Its volume is 72 cm cubed.

A mathematically similar prism has volume 1125 cm cubed.

Work out the total length of all the edges of the larger prism.

The misconception scales the edge length by the volume factor 15.625. The fix: volume factor $= 1125 / 72 = 15.625$ , so the linear factor $= \sqrt[3]{15.625} = 2.5$ , and the scaled edge total $= 60 \times 2.5 = 150$ cm.

Why students fall for this

Forward scaling is taught and practised far more than the reverse. Students learn to cube a length factor to scale a volume, but rarely meet the question that hands them a volume ratio and asks for the length, so the cube root never becomes automatic.

Reversing feels like dividing. Because cubing is repeated multiplication, undoing it feels like it should be a division, and dividing the volume factor by 3 is a tempting but wrong shortcut. The correct inverse of a cube is a cube root, not a division by 3.

One ratio gets stretched over everything. Once a student has a ratio in hand, it is easy to reuse it for whatever quantity the question asks about next, ignoring that length, area and volume sit at different powers, and ignoring whether the shapes are even similar.

The fix: Reverse with a root, and never reuse one ratio as another

Step 1: name which quantity you are given and which you want. Volume to length, area to length, or one ratio to a different one. The direction decides the operation.

Step 2: reverse with the matching root. Volume to length is a cube root: $\sqrt[3]{8} = 2$ , or $\sqrt[3]{15.625} = 2.5$ . Area to length is a square root: $\sqrt{9} : \sqrt{16} = 3 : 4$ .

Step 3: apply the linear factor to the length. Only the linear factor may scale a length: scaled edge total $= 60 \times 2.5 = 150$ cm, never $60 \times 15.625$ .

Step 4: never copy a ratio across dimensions. Compute each quantity from the actual dimensions. Cube A and cuboid B have a volume ratio of 1 to 2 but a surface area ratio of $6 : 10 = 3 : 5$ , and they are not even similar.

Worked example

JUN24·2·25 structure: prism volume 72 cm cubed, similar prism volume 1125 cm cubed, original edges total 60 cm. Find the scaled total edge length.

Volume scale factor. $1125 / 72 = 15.625$ .
Cube root to the linear factor.
$\sqrt[3]{15.625} = 2.5 \quad (2.5^3 = 15.625)$
Scale the length. $60 \times 2.5 = 150$ cm. Trap: $60 \times 15.625 = 937.5$ , scaling a length by the volume factor.

JUN23·3·6 structure: cube A is 1 by 1 by 1, cuboid B is 1 by 1 by 2. The volume ratio is 1 to 2. Is the surface area ratio also 1 to 2?

Surface area of A. Six 1 by 1 faces: $6$ .
Surface area of B. Two 1 by 1 ends and four 1 by 2 sides: $2 + 8 = 10$ .
Form the ratio. $6 : 10 = 3 : 5$ , not the 1 to 2 volume ratio. A and B are not similar, so no single scale factor exists.

Constructed area-to-length example: two similar shapes have areas in the ratio 9 to 16. Find the length ratio. (This is a built example to drill the reverse move; the same square root appears in proportion questions such as $G \propto \sqrt{H}$ .)

Area ratio is the length ratio squared. So square-root each part.
Square root.
$\sqrt{9} : \sqrt{16} = 3 : 4$
Check forward. $3^2 : 4^2 = 9 : 16$ , which matches. Trap: leaving it as 9 to 16.

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Common questions

How do you get a length ratio from a volume ratio?: Take the cube root. The volume ratio is the length ratio cubed, so $\sqrt[3]{8} = 2$ and $\sqrt[3]{27} = 3$ . Dividing the volume ratio by 3 is the common wrong move.
A prism of volume 72 cm cubed scales to a similar prism of 1125 cm cubed. What is the linear factor?: $\sqrt[3]{1125 / 72} = \sqrt[3]{15.625} = 2.5$ . A 60 cm edge total then scales to $60 \times 2.5 = 150$ cm, not $937.5$ .
Why is the surface area ratio of a 1 by 1 by 1 cube to a 1 by 1 by 2 cuboid not 1 to 2?: Because surface area and volume are different dimensions, and the two solids are not similar. Work the areas out directly: $6$ and $10$ , so the ratio is $3 : 5$ , while the volumes are in the ratio 1 to 2.

Related misconceptions

Area scales by the squareWhen lengths scale by k, areas scale by k squared, not k. A radius ratio of 4 to 1 is an area ratio of 16 to 1, and a length factor of 0.4 is an area factor of 0.16.
Volume scales by the cubeWhen lengths scale by k, volumes scale by k cubed, not k. A linear factor of 12 is a volume ratio of 1728 to 1, and a cylinder with radius 3r and height 2h holds 18 times as much.

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