If two figures are similar with length scale factor k (so every matching length in the second figure is k times the first), then the area scale factor is k² and the volume scale factor is k³. In ratio form, A₁ : A₂ = l₁² : l₂² and V₁ : V₂ = l₁³ : l₂³. Never use k itself for area or volume: that is the single most common mistake in this topic.
The idea of similar figures feels simple: bigger version, same shape. So it is tempting to assume that if the sides are twice as long, the area is also twice as big, and the volume is also twice as big. It is not. Area is a two-dimensional measurement built from two lengths multiplied together, so doubling every length quadruples the area. Volume is built from three lengths multiplied together, so doubling every length gives eight times the volume.
This single idea, that area scales as the square of the length ratio and volume scales as the cube of the length ratio, covers a large share of the "similar figures" and "similar solids" questions in O-Level E-Maths, including mass and capacity questions dressed up in real-world language. It is also exactly why, on ratio and map scales questions, an area on a map is scaled by the square of the length scale factor, not the scale factor itself.
Step 1
Find the length scale factor k = l₁ / l₂ from a matching pair of sides, heights, or radii
Step 2
For area: square k. Area ratio = k²
Step 3
For volume: cube k. Volume ratio = k³
Step 4
Multiply the known area or volume by the correct scale factor to get the unknown one
In full ratio form, for two similar figures with corresponding lengths l₁ and l₂, corresponding areas A₁ and A₂, and corresponding volumes V₁ and V₂:
A₁ / A₂ = (l₁ / l₂)² and V₁ / V₂ = (l₁ / l₂)³
The same rule applies to mass, when two objects are made of the same material at the same density: mass scales exactly like volume, so M₁ / M₂ = (l₁ / l₂)³ as well.
Triangles ABC and PQR are similar, with AB = 3 cm and the corresponding side PQ = 5 cm. The area of triangle ABC is 18 cm². Find the area of triangle PQR.
Solution
ABC to PQR using the matching sides: k = PQ / AB = 5 / 3.k² = (5/3)² = 25/9.Area PQR = 18 × 25/9.18 ÷ 9 = 2, so Area PQR = 2 × 25 = 50 cm².Quick check: since PQR is the larger triangle, its area must be bigger than 18 cm², and 50 cm² is bigger. If your answer for the larger figure comes out smaller, you have inverted the ratio, use k = larger / smaller, never the other way round.
Two similar cylindrical containers have heights 8 cm and 12 cm. The smaller container holds 128 cm³ of water when full. Find the capacity of the larger container.
Solution
k = 12 / 8 = 3/2.k³ = (3/2)³ = 27/8.Capacity larger = 128 × 27/8.128 ÷ 8 = 16, so Capacity larger = 16 × 27 = 432 cm³.Same idea for mass: if the two containers were made of the same solid material, a mass question would use the same working, cube the length scale factor, because mass follows volume for objects of the same material and density.
The step students get wrong
Using k itself, instead of k² or k³, when scaling an area or a volume. If a question gives a length ratio of 2 : 3 and asks for the area ratio, the answer is 4 : 9, not 2 : 3. For a volume ratio, it is 8 : 27. Before you multiply, ask yourself what you are scaling: a length stays as k, an area needs k², a volume needs k³. Write this down as a mini check on every question in this topic.
How do I know whether to square or cube the scale factor?
Count the dimensions of what you are measuring. A length has one dimension, so it uses k. An area has two dimensions, so it uses k². A volume has three dimensions, so it uses k³. This also works for mass, since mass of similar solids of the same material follows volume, so it uses k³ too.
What if I am only given areas, and need to find a length ratio?
Work backwards: take the square root of the area ratio to get the length ratio. For example, if two similar figures have areas in the ratio 16 : 25, the length ratio is √16 : √25 = 4 : 5. The same idea applies to volumes: take the cube root of the volume ratio to recover the length ratio.
Does this work for any similar shapes, not just triangles and cylinders?
Yes. The k, k², k³ relationship holds for any pair of similar figures or similar solids, regardless of the shape, as long as they are genuinely similar (same shape, corresponding angles equal, corresponding lengths in the same ratio). It applies to circles, polygons, cones, spheres, and composite solids.
How is this different from the congruence and similarity tests?
The congruence tests (SSS, SAS, AAS, RHS) and similarity tests (AA, SSS, SAS) are used to prove two figures are congruent or similar in the first place. Area and volume scale factors are what you calculate afterwards, once similarity has already been established or given in the question.
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