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

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Practice 99 Solid geometry (surface area and volume) questions in the app

What this topic is

Solid geometry (surface area and volume) covers prisms, cubes, cylinders and cones.

Formulas are often supplied in the question, so the real test is applying them and comparing results. Changing the radius of a cylinder affects volume more than changing its height, because the radius is squared: doubling the radius makes the volume four times larger. A cone holds one-third the volume of a cylinder with the same base and height.

Painted-cube puzzles appear too, asking how many small cubes have paint on a certain number of faces; corners, edges and face centers must be counted separately. Quantitative comparisons may set two solids side by side.

Sample questions

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

Question 1Easier

The volume of a cylinder is $π r^{2} h$ . Cylinder $P$ has radius 2 and height 9. Cylinder $Q$ has radius 3 and height 4.

Column A	Column B
The volume of Cylinder $P$	The volume of Cylinder $Q$

The quantity in Column A is greater✗ Not this oneThe quantity in Column B is greater✗ Not this oneThe two quantities are equal✓ CorrectThe relationship cannot be determined from the information given✗ Not this one

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Answer: C — The two quantities are equal

Cylinder $P$ has volume $π (2)^{2} (9) = 36 π$ , and Cylinder $Q$ has volume $π (3)^{2} (4) = 36 π$ . The two quantities are equal.

Question 2Mid

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?

8✗ Not this one24✓ Correct32✗ Not this one36✗ Not this one

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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 $4 - 2 = 2$ such cubes, so there are $12 \times 2 = 24$ .

Question 3Easier

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?

\frac{1}{8}

✗ Not this one

\frac{1}{3}

✓ Correct

\frac{3}{8}

✗ Not this one

\frac{1}{2}

✗ Not this one

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Answer: B — $\frac{1}{3}$

The volume of a cone is $\frac{1}{3} π r^{2} h$ and the volume of a cylinder is $π r^{2} h$ . With equal radius and height, $\frac{\frac{1}{3} π r^{2} h}{π r^{2} h} = \frac{1}{3}$ .

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 99 Solid geometry (surface area and volume) questions in the app

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

What this topic is

Sample questions

More ISEE Upper topics

Quantitative Reasoning