If the length scale factor is k, what is the volume scale factor?

The volume scale factor is k cubed. Volume is three lengths multiplied together, and each length carries the factor k, so the volume carries k times k times k. Doubling every length makes the volume 8 times bigger, not 2. Tripling every length makes the volume 27 times bigger, not 3.

Two similar cones have radii 2 cm and 24 cm. How many times bigger is the large cone's volume?

1728 times. The linear scale factor is 24 over 2 = 12, and volume uses the cube of the linear factor, so the volume scale factor is 12 cubed = 1728. The trap, 12, leaves the factor un-cubed, and 144 squares it as if it were an area.

A small paving slab (side 20 cm) is similar to a large slab (side 25 cm) costing 17.50 pounds. What does the small slab cost?

8.96 pounds. Cost follows the material, which is a volume, so it scales by the cube of the linear factor. The factor from larger to smaller is 20 over 25 = 0.8, and 0.8 cubed = 0.512, so the small slab costs 17.50 times 0.512 = 8.96 pounds. The trap, 14.00 pounds, leaves 0.8 un-cubed; squaring it gives 11.20 pounds.

GCSE Maths Higher · AQA · Similar shapes & scaling

Volume scales by the cube of the length factor

When two solids are similar, every length is multiplied by the same length scale factor k. But volume is three lengths multiplied together, so the volume scale factor is $k^3$ , not k. Double every length and the solid holds 8 times as much; triple it and it holds 27 times as much. Applying the length factor straight to a volume, the illusion of linearity, is the single biggest grade 7 to 9 mark-loser in similar-solids questions.

The Higher trap bites hardest with ratio and stretched scale factors. A linear factor of 12 is a volume ratio of 1728 to 1. A linear factor of 0.8 is a volume factor of 0.512. And when a radius and a height scale by different amounts, there is no single factor to cube: a cylinder enlarged to radius 3r and height 2h holds 18 times as much, because the radius enters the formula squared.

Already know this is your gap? Skip the diagnostic and jump straight into the targeted lesson.

How to spot it in your own work

You multiplied a volume by the length scale factor k, when you should have multiplied by $k^3$ .
You squared the linear factor instead of cubing it, using the area rule $k^2$ for a volume.
With a fractional linear factor like 0.8, you forgot that the volume factor 0.512 is smaller, and your scaled cost or volume came out too large.
When a radius and a height scaled by different factors, you multiplied them directly instead of substituting into $V = \pi r^2 h$ , missing the square on the radius.

An exam question that triggers it

Here is the structure of JUN22 Paper 2 Question 13b:

Two cones are mathematically similar.

The small cone has base radius 2 cm. The large cone has base radius 24 cm.

How many times bigger is the volume of the large cone than the small cone?

The misconception reads the linear factor 12 straight off as the volume factor, or squares it to 144. The fix: linear factor $= 24 / 2 = 12$ , so the volume factor $= 12^3 = 1728$ .

Why students fall for this

Linear scaling is met first and overlearned: double the price, double the distance, double the recipe. The brain reaches for proportional thinking by default, and volume feels like just another quantity to scale by the same factor.

The cube is invisible in the wording. A question gives a linear factor or a pair of radii, and nothing on the page shouts that the volume ratio is the cube. The candidate has to supply the cube from the structure of volume itself, that it is length times length times length.

Mixed-up powers make it worse. Students who half-remember that similar shapes involve powers reach for the square, the area rule, and stop there: they answer 144 instead of 1728, or 11.20 pounds instead of 8.96, squaring when they should cube.

The fix: Find the linear factor, cube it, then scale the volume

Step 1: find the linear scale factor k. Use a pair of corresponding lengths: $k = \tfrac{24}{2} = 12$ , or $\tfrac{20}{25} = 0.8$ .

Step 2: cube it for the volume scale factor. $k^3 = 12^3 = 1728$ , or $0.8^3 = 0.512$ . Do this before touching any volume or cost.

Step 3: scale the volume or cost in the right direction. Going from larger to smaller you multiply by the cube of a fraction below 1, so the answer shrinks: the smaller slab costs $17.50 \times 0.512 = 8.96$ pounds.

Step 4: when dimensions stretch by different factors, use the formula. Substitute into $V = \pi r^2 h$ . The radius is squared, so a radius factor of 3 becomes 9, and a cylinder with radius 3r and height 2h scales by $3^2 \times 2 = 18$ .

Worked example

JUN22·2·13b structure: similar cones, small base radius 2 cm, large base radius 24 cm. How many times bigger is the large cone's volume?

Linear factor. $24 / 2 = 12$ .
Cube it.
$12^3 = 12 \times 12 \times 12 = 1728$
Read off. The large cone's volume is $1728$ times bigger. Trap: $12$ leaves it un-cubed, and $144$ squares it as an area.

NOV23·2·25 structure: similar paving slabs, sides 20 cm and 25 cm, the larger costs 17.50 pounds. Find the cost of the smaller.

Linear factor (larger to smaller). $20 / 25 = 0.8 = \tfrac{4}{5}$ .
Cube it. $\left(\tfrac{4}{5}\right)^3 = \tfrac{64}{125} = 0.512$ .
Scale the cost. $17.50 \times 0.512 = 8.96$ pounds. Traps: $17.50 \times 0.8 = 14.00$ (un-cubed) and $17.50 \times 0.64 = 11.20$ (squared).

JUN24·3·21b structure: a cylinder of radius r and height h is enlarged to radius 3r and height 2h. How many times bigger is its volume?

Substitute into the formula.
$\pi (3r)^2 (2h) = \pi \cdot 9r^2 \cdot 2h = 18 \, \pi r^2 h$
Read off the factor. $3^2 \times 2 = 18$ times bigger.
Apply it. A small volume of 50 cubic cm becomes $50 \times 18 = 900$ cubic cm. Trap: $3 \times 2 = 6$ gives only $300$ , ignoring the square on the radius.

Find out if this is costing you marks

The 10-minute diagnostic checks for this pattern (and four others) using AQA-style GCSE Higher items. Free, no signup, anonymous.

Take the diagnostic →

Common questions

If lengths scale by k, what does the volume scale by?: $k^3$ . Volume is length times length times length, and each length carries the factor k, so the volume carries $k \times k \times k = k^3$ . Doubling lengths gives 8 times the volume; tripling gives 27 times.
Two similar cones have radii 2 cm and 24 cm. How many times bigger is the large cone's volume?: $1728$ . The linear factor is $24 / 2 = 12$ , and volume uses the cube, so $12^3 = 1728$ , not 12 and not 144.
Why does a cylinder with radius 3r and height 2h scale by 18, not 6?: Because volume is $\pi r^2 h$ , so the radius enters squared. Its factor of 3 becomes $3^2 = 9$ , and the height factor 2 stays 2, giving $9 \times 2 = 18$ . Multiplying the stretches directly gives the wrong answer 6.

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.
Reverse and cross-dimension scalingTo recover a length from a volume you cube root, not divide; and area and volume never share the same ratio because one squares the length factor and the other cubes it.

← All GCSE Maths Higher misconceptions