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Solid geometry (surface area and volume) — ISEE Upper practice questions

On the ISEE Upper Quantitative Reasoning section, Solid geometry (surface area and volume) covers cubes, cylinders, cones, and other three-dimensional figures, including volume comparisons and painted faces. A student must apply given formulas, keep radius and height distinct, and tell surface area from volume. Questions often place two cylinders with different radii and heights side by side, ask what fraction a cone's volume is of a cylinder with the same radius and height, or count unit cubes with paint on exactly two faces after a cube is cut apart. Traps include swapping dimensions and treating corners as edges. Choices are nearby values, fractions, or comparison options.

  • Section: Quantitative Reasoning 37 questions on the paper
  • 98 practice questions in the app

Sample questions

Question 1

The volume of a cylinder is πr2h\pi r^2h. Cylinder AA has radius 2 and height 9. Cylinder BB has radius 3 and height 4.

Column AColumn B
The volume of Cylinder AAThe volume of Cylinder BB
  1. AThe quantity in Column A is greater
  2. BThe quantity in Column B is greater
  3. CThe two quantities are equal
  4. DThe relationship cannot be determined from the information given
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Answer: C — The two quantities are equal

Cylinder AA has volume π(2)2(9)=36π\pi(2)^2(9)=36\pi, and Cylinder BB has volume π(3)2(4)=36π\pi(3)^2(4)=36\pi. The volumes are equal.

Question 2

A cube measuring 4 units on each edge is painted on all six faces and then cut into unit cubes. How many of the unit cubes have paint on exactly two faces?

  1. A8
  2. B24
  3. C32
  4. D36
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Answer: B — 24

Exactly-two-face cubes lie along the 12 edges, excluding the two corner cubes on each edge. Each edge contributes 4−2=24-2=2 such cubes, so there are 12×2=2412\times2=24.

Question 3

A cone has radius 3 centimeters and height 8 centimeters. A cylinder has the same radius and the same height. The volume of the cone is what fraction of the volume of the cylinder?

  1. A18\dfrac{1}{8}
  2. B13\dfrac{1}{3}
  3. C38\dfrac{3}{8}
  4. D12\dfrac{1}{2}
Show answer

Answer: B — 13\dfrac{1}{3}

The volume of a cone is 13πr2h\dfrac{1}{3}\pi r^2 h and the volume of a cylinder is πr2h\pi r^2 h. With equal radius and height, 13πr2hπr2h=13\dfrac{\dfrac{1}{3}\pi r^2 h}{\pi r^2 h}=\dfrac{1}{3}.

Practice 98 Solid geometry (surface area and volume) questions in the app

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