GRE and Grad School Admissions Blog

Geometry Challenge: Use the Volume Formula Wisely

GRE Geometry Challenge (Pictured: Skewb, a mechanical puzzle)

GRE geometry can sometimes seem like a formula fest. Formulas matter, for sure, but simply memorizing them isn’t enough. You also need to know how to use them  efficiently. For practice, try this Quantitative Comparison question that requires you to apply the volume formula for a cylinder.

Geometry: 3-D Figures
Difficulty Level: 3 (Medium)

Question Difficulty
  • Very High50
  • High420
  • Medium340
  • Low26080
  • Very Low1100
% of Test Takers Who Answered Correctly

In The Practice Book for the Paper-based GRE, a moderate-difficulty Quant question gives you the radius and volume of a right circular cylinder and asks you to compare the cylinder’s height, which is not given, to a known number of inches. (See §6, 7.) Just 46% of examinees got the question right when it was on a real exam. Here’s a similar problem that starts you off the with the cylinder’s height and volume and asks you to think about the size of its radius.

THE CHALLENGE

A right circular cylinder with height 2 inches has volume 16 cubic inches.

  • Quantity A
  • The radius of the cylinder
  • Quantity B
  • 4 inches
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

⇓SOLVED⇓

The Solution

Resist the urge to solve for quantity A. Instead, see whether A, the cylinder’s radius, could be equal to B, 4 inches. If a radius of 4 and a height of 2 don’t yield a volume of 16, then A cannot be equal to B.

Volume of a Right Circular Cylinder = πr2h

  • Let r = 4 and h = 2.
  • π(4)2(2) > 16
  • Thus, r < 4 so (B) is correct.

With radius 4, the cylinder’s volume is greater than 16. So 4 must be greater than the cylinder’s radius.

For this simple solution, you need to know the volume formula for a right circular cylinder. This formula fits a more basic volume formula that you should memorize for the exam.

GRE Volume Formula   Area of Base × Height

This “master” formula works for right circular cylinders plus rectangular solids, cubes, and other prisms. If you’re asked to find the volume of other 3-D figures, such as cones, spheres, or pyramids, the question will give you the required formula.

Knowing the right formula is one thing; using it wisely is another. In the Challenge problem, solving for the cylinder’s radius would’ve just slowed you down. You would’ve had to isolate r and then find its exact value with your calculator.

  • πr2(2) = 16
  • r2 = 16 ÷ (2π)
  • r = √[16 ÷ (2π)]

You avoid algebra and any difficult calculations by substituting 4, quantity B, for the cylinder’s radius, quantity A. What’s more, you get right to the task that gives name to the question type: Quantitative Comparison.

Ready for another GRE Quant Challenge? Check out this post: GRE Arithmetic Challenge: Find the Expression that Must Be Negative.

Photo: “Skewb“. Colors modified. Licensed under CC BY-SA 3.0 via Commons.

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