Solid geometry (volume) — SSAT Upper practice questions

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Practice 96 Solid geometry (volume) questions in the app

What this topic is

Solid geometry (volume) on the SSAT Upper Quantitative section covers the volumes of solids such as rectangular prisms and cylinders, often linked to surface area or a percent change in dimensions.

A student must apply volume formulas, relate a height written in terms of a base edge, factor integer side lengths that fit a given volume, and scale radius and height correctly. Questions are short word problems whose five answer choices are numbers or percents.

Traps include swapping volume and surface area, treating a change in radius as linear rather than squared, and reporting one leftover edge instead of the sum or measure asked.

Sample questions

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Question 1Harder

A right rectangular prism has a square base with side length $s$ inches. The height of the prism is $3s$ 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 $s\cdot s\cdot3s=3s^3$ . Since $3s^3=192$ , $s=4$ and the height is 12. The surface area is $2(4\times4)+4(4\times12)=224$ .

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 $5ab=180$ , so $ab=36$ . The surface area gives $2(5a+5b+ab)=202$ , so $5a+5b+36=101$ . 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

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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 $2.25\times0.8=1.8$ , an $80\%$ increase.

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Practice 96 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)