Question type

Area and volume of similar shapes on the SAT

For similar figures with length scale factor $k$ , lengths scale by $k$ , areas and surface areas by $k^{2}$ , and volumes, and so masses, by $k^{3}$ . If edge lengths are in the ratio $2 : 5$ , volumes are in the ratio $8 : 125$ . To go from an area ratio to a volume ratio, take the square root, then cube.

These questions give two similar solids or shapes and one ratio, of lengths, areas or volumes, and ask for another measure.

It appeared on one of the five official practice tests we analysed. When it is missed, the cause is almost always the wrong power.

On 1 of 5 official practice tests
Desmos helps

Why students miss it

Scaling volume by the length ratio. Prisms with edges in the ratio $2 : 5$ have volumes in the ratio $8 : 125$ , not $2 : 5$ .

Mass is volume. A model one quarter the height of a solid statue weighs $\frac{1}{64}$ of it.

Starting from an area ratio and cubing it. Take the square root first to get $k$ .

How to spot it

"Similar" solids, a scale model, or every edge "tripled".
A ratio of one measure and a question about another: heights given, volume asked.
A percent increase in a radius with a question about volume.

The method

Find k, then use the right power

Get the length scale factor $k$ from whatever ratio you are given.

Lengths, perimeters: times $k$ .
Areas, surface areas: times $k^{2}$ .
Volumes, capacities, masses: times $k^{3}$ .
Given an area ratio: $k$ is its square root.

With Desmos

Desmos gives the exact power quickly.

For edges in the ratio 2 to 5: type 48*(5/2)^3 and read 750.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#683Geo

Two similar rectangular prisms have corresponding edge lengths in the ratio $2 : 5$ . The volume of the smaller prism is 48 cubic centimeters. What is the volume, in cubic centimeters, of the larger prism?

Pick an answer, then check it.

Show the answer and explanation

Volumes of similar solids scale as the cube of the edge ratio:

(2 : 5)^{3} = 8 : 125

. So

\frac{48}{V} = \frac{8}{125}

, giving

V = 48 \cdot \frac{125}{8} = 750

cubic centimeters.

2Question 2 of 2#843Geo

Two similar right circular cylinders have surface areas in the ratio 9 to 25. The volume of the smaller cylinder is 54 cubic centimeters. What is the volume, in cubic centimeters, of the larger cylinder?

Pick an answer, then check it.

Show the answer and explanation

Surface areas scale as the square of the linear ratio, so

9 : 25

means lengths are in ratio

3 : 5

. Volumes scale as the cube:

27 : 125

. Then

54 \cdot \frac{125}{27} = 250

cubic centimeters.

Questions students ask

If the edges of a cube are tripled, what happens to the volume?

It is multiplied by 3³ = 27. Every one of the three dimensions triples.

How do surface area and volume scale for similar solids?

Surface area scales by the square of the length ratio and volume by its cube.

Practise this type until it is automatic

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