Question type

Volume from surface area and other indirect dimensions

When an SAT volume question does not give the dimension you need, find it first from what is given, then use the volume formula. A cube's surface area $6 s^{2} = 384$ gives $s = 8$ and volume 512. A pyramid's volume $\frac{1}{3} s^{2} h = 400$ with $h = 12$ gives $s = 10$ .

These questions give a surface area, a circumference, a base area or a volume, and ask for a different measure that needs a dimension you have to work out first.

It appeared on one of the five official practice tests we analysed. The reference sheet gives the formulas; the question tests whether you can run them backwards.

On 1 of 5 official practice tests
Desmos helps

Why students miss it

Plugging the given number straight into the volume formula as if it were the length.

Forgetting the $\frac{1}{3}$ for pyramids and cones.

Stopping at the dimension: the question usually asks for the volume, area or another length after it.

How to spot it

A volume or area question where one key dimension is not stated.
A surface area, circumference or base area given instead.
Cubes, spheres, pyramids and cones, whose formulas are on the reference sheet.

The method

Dimension first

Write the formula for what is GIVEN and solve it for the missing dimension.

Cube surface area: $6 s^{2}$ , so $s = SA /6$ .
Circumference: $r = \frac{C}{2 π}$ .
Pyramid or cone volume: remember the $\frac{1}{3}$ .
Then: use the dimension in the formula you were asked about.

With Desmos

Desmos solves the formula for the missing dimension.

Type 6s^2=384 and read s = 8, then type 8^3 for the volume.

Two real questions

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

1Question 1 of 2#27Geo

A cube has a surface area of 384 square centimeters. What is the volume, in cubic centimeters, of the cube?

Pick an answer, then check it.

Show the answer and explanation

A cube has 6 equal faces, so

6 s^{2} = 384

, giving

s^{2} = 64

and

s = 8

. Then volume

= 8^{3} = 512

cubic centimeters.

2Question 2 of 2#854Geo

A right pyramid with a square base has a volume of 400 cubic feet and a height of 12 feet. What is the side length, in feet, of the base?

Pick an answer, then check it.

Show the answer and explanation

For a pyramid,

V = \frac{1}{3} (base area) h

. So

400 = \frac{1}{3} s^{2} (12) = 4 s^{2}

, giving

s^{2} = 100

and

s = 10

feet.

Questions students ask

How do you find the volume of a cube from its surface area?

Divide the surface area by 6 to get one face, take the square root to get the edge, then cube it. 384 ÷ 6 = 64, edge 8, volume 512.

Are volume formulas given on the SAT?

Yes, the reference sheet lists the volume formulas for prisms, cylinders, spheres, cones and pyramids.

Practise this type until it is automatic

Five more indirect-volume questions, checked instantly, with every wrong choice explained. Free, no card.

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