Question type

Hollow cylinder volume on the SAT

The volume of the material in a hollow cylinder, such as a pipe, is the outer cylinder minus the inner one: $V = π (R^{2} - r^{2}) h$ . With outer radius 5, inner radius 3 and length 12, that is $π (25 - 9) (12) = 192 π$ . Wall thickness is $R - r$ .

These questions describe a pipe, tube or culvert and ask for the volume of the material it is made of, or work back from that volume to a missing radius, height or wall thickness.

It appeared on one of the five official practice tests we analysed, twice.

On 1 of 5 official practice tests
Desmos helps

Why students miss it

Subtracting the radii before squaring. $π (5 - 3)^{2} (12)$ is not the same as $π (5^{2} - 3^{2}) (12)$ .

Using the outer volume alone, as if the pipe were solid.

Wall thickness: a 5 cm outer radius with a 1 cm wall has an inner radius of 4, not 6.

How to spot it

"Hollow", "pipe", "tube", "culvert", "the material that forms".
Two radii, or one radius and a wall thickness.

The method

Outer minus inner

Find both radii first.

Volume of the material: $π R^{2} h - π r^{2} h$ .
Wall thickness $t$ given: inner radius $= R - t$ .
Volume given: solve $π (R^{2} - r^{2}) h = V$ for the missing value.

With Desmos

Desmos solves for a missing radius once the equation is written.

Type pi(25-r^2)12=192pi and read r = 3, so the wall is 2 cm thick.

Two real questions

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

1Question 1 of 2#31Geo

A section of pipe is a hollow right circular cylinder with an outer radius of 5 centimeters, an inner radius of 3 centimeters, and a length of 12 centimeters. What is the volume, in cubic centimeters, of the material that forms the pipe?

Pick an answer, then check it.

Show the answer and explanation

The volume of the material is the outer cylinder minus the hollow inner cylinder:

V = π R^{2} h - π r^{2} h = π (5^{2} - 3^{2}) (12) = π (25 - 9) (12) = π (16) (12) = 192 π

cubic centimeters.

2Question 2 of 2#685Geo

A metal tube is a hollow right circular cylinder with length $12$ centimeters and outer radius $5$ centimeters. The volume of the metal forming the wall of the tube is $192 π$ cubic centimeters. What is the thickness, in centimeters, of the wall?

Pick an answer, then check it.

Show the answer and explanation

Call the inner radius

r

. The wall volume is

π (5^{2} - r^{2}) (12) = 192 π

. Divide both sides by

12 π

to get

25 - r^{2} = 16

, so

r^{2} = 9

and

r = 3

. The wall thickness is the outer radius minus the inner radius:

5 - 3 = 2

centimeters.

Questions students ask

What is the formula for the volume of a hollow cylinder?

π(R² − r²)h, where R is the outer radius, r the inner radius and h the height or length.

Is the hollow cylinder formula on the SAT reference sheet?

The cylinder formula V = πr²h is. The hollow version is that formula used twice and subtracted.

Practise this type until it is automatic

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