Solid geometry (volume) — SSAT Upper practice questions

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Practice 103 Solid geometry (volume) questions in the app

What this topic is

Solid geometry (volume) deals with three-dimensional shapes, usually rectangular prisms, cubes and cylinders.

The volume of a prism is length times width times height, while a cylinder's volume is pi times the radius squared times the height. Because the radius is squared, a change to it counts twice: doubling the radius of a cylinder makes its volume four times as large, not twice.

SSAT Upper items frequently pair volume with surface area, which adds the areas of all faces, or set up an equation where a side length must be found from a given volume. Confusing surface area with volume, or forgetting to square the radius, produces several wrong choices.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Harder

A right rectangular prism has a square base with side length $s$ inches. The height of the prism is $3 s$ inches, and the volume of the prism is 192 cubic inches. What is the surface area of the prism, in square inches?

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Answer: C — 224

The volume is $s \cdot s \cdot 3 s = 3 s^{3}$ . Since $3 s^{3} = 192$ , $s = 4$ and the height is 12. The surface area is $2 (4 \times 4) + 4 (4 \times 12) = 224$ .

Question 2Harder

A rectangular prism has integer side lengths, volume 180 cubic units, and one side length of 5 units. If the surface area of the prism is 202 square units, what is the sum of the other two side lengths?

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Answer: D — 13

Let the side lengths be 5, $a$ , and $b$ . The volume gives $5 a b = 180$ , so $a b = 36$ . The surface area gives $2 (5 a + 5 b + a b) = 202$ , so $5 a + 5 b + 36 = 101$ . Therefore $5 (a + b) = 65$ , and $a + b = 13$ .

Question 3Harder

The radius of a cylinder is increased by $50 %$ , and the height of the cylinder is decreased by $20 %$ . By what percent does the volume of the cylinder change?

20 %

decrease✗ Not this one

20 %

increase✗ Not this one

50 %

increase✗ Not this one

80 %

increase✓ Correct

125 %

increase✗ Not this one

Show answer

Answer: D — $80 %$ increase

The radius is multiplied by 1.5, so $r^{2}$ is multiplied by $(1.5)^{2} = 2.25$ . The height is multiplied by 0.8, making the volume factor $2.25 \times 0.8 = 1.8$ , an $80 %$ increase.

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Practice 103 Solid geometry (volume) questions in the app

Solid geometry (volume) — SSAT Upper practice questions

What this topic is

Sample questions

More SSAT Upper topics

Quantitative (Math)