Study on the flySSAT

Study on the fly›SSAT Upper›Quantitative (Math) › Solid geometry (volume)

Quantitative (Math)Free

Solid geometry (volume) — SSAT Upper practice questions

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.

  • Section: Quantitative (Math) 50 questions on the paper
  • 91 practice questions in the app

Sample questions

Question 1

A right rectangular prism has a square base with side length ss inches. The height of the prism is 3s3s inches, and the volume of the prism is 192 cubic inches. What is the surface area of the prism, in square inches?

  1. A192
  2. B208
  3. C224
  4. D240
  5. E256
Show answer

Answer: C — 224

The volume is s⋅s⋅3s=3s3s\cdot s\cdot3s=3s^3. Since 3s3=1923s^3=192, s=4s=4 and the height is 12. The surface area is 2(4×4)+4(4×12)=2242(4\times4)+4(4\times12)=224.

Question 2

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?

  1. A9
  2. B10
  3. C12
  4. D13
  5. E18
Show answer

Answer: D — 13

Let the side lengths be 5, aa, and bb. The volume gives 5ab=1805ab=180, so ab=36ab=36. The surface area gives 2(5a+5b+ab)=2022(5a+5b+ab)=202, so 5a+5b+36=1015a+5b+36=101. Therefore 5(a+b)=655(a+b)=65, and a+b=13a+b=13.

Question 3

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

  1. A20%20\% decrease
  2. B20%20\% increase
  3. C50%50\% increase
  4. D80%80\% increase
  5. E125%125\% increase
Show answer

Answer: D — 80%80\% increase

The radius is multiplied by 1.5, so r2r^2 is multiplied by (1.5)2=2.25(1.5)^2=2.25. The height is multiplied by 0.8, making the volume factor 2.25×0.8=1.82.25\times0.8=1.8, an 80%80\% increase.

Practice 91 Solid geometry (volume) questions in the app

More SSAT Upper topics

Quantitative (Math)

Everything on the SSAT Upper

Study on the fly is an independent study tool and is not affiliated with, sponsored by, or endorsed by any testing organization, school, or government body.

SSAT® is a registered trademark of The Enrollment Management Association.

All practice questions and passages are original works created for Study on the fly and are not reproduced from any official test.