GRE Volume Questions Practice Questions with Answers

Volume measures the three-dimensional space occupied by an object and is a frequent topic on the GRE Quantitative Reasoning section. Calculating the capacity of solids like cubes, rectangular prisms, and cylinders requires a firm grasp of geometric formulas and the ability to manipulate algebraic variables. Success on GRE Volume Questions often depends on your speed in identifying which formula to apply and how to handle units of measurement effectively.

For students aiming for a high score, integrating these geometry concepts into a broader GRE Prep strategy is essential. Whether you are comparing the volumes of two different shapes in a Quantitative Comparison question or solving for a missing dimension in a word problem, precision is key. You can further refine your skills using an AI Question Generator to create custom drills based on the specific shapes you find most challenging.

Concept Explanation

GRE Volume Questions test your ability to calculate the space inside three-dimensional figures using standardized geometric formulas. Most questions on the GRE focus on three primary solids: rectangular solids (including cubes), right circular cylinders, and occasionally spheres or cones. The volume is always expressed in cubic units, such as $\text{cm}^3$ or $\text{in}^3$.

To solve these problems, you must memorize the following core formulas:

• Rectangular Solid: $V = l \times w \times h$ (length $\times$ width $\times$ height)
• Cube: $V = s^3$ (where $s$ is the side length)
• Right Circular Cylinder: $V = \pi r^2 h$ (area of the circular base $\times$ height)

Beyond these formulas, the GRE often tests the relationship between surface area and volume. For instance, increasing the side of a cube by a factor of 2 increases its surface area by a factor of 4 ($2^2$) but increases its volume by a factor of 8 ($2^3$). Understanding these scaling factors is a common shortcut for difficult problems. According to Wikipedia's entry on Volume, the concept is fundamental to both pure mathematics and applied sciences, making it a staple of standardized testing.

Solved Examples

1. Problem: A rectangular tank has a length of 10 cm, a width of 5 cm, and is filled with water to a height of 4 cm. If a solid metal cube with a side length of 2 cm is dropped into the tank and sinks to the bottom, what is the new height of the water?
   1. Calculate the initial volume of water: $V_{ \text{water}} = 10 \times 5 \times 4 = 200 \text{ cm}^3$.
   2. Calculate the volume of the submerged cube: $V_{ \text{cube}} = 2^3 = 8 \text{ cm}^3$.
   3. Add the volumes to find the total volume: $200 + 8 = 208 \text{ cm}^3$.
   4. The base area of the tank remains $10 \times 5 = 50 \text{ cm}^2$. Solve for the new height $h$: $50 \times h = 208$.
   5. $h = \frac{208}{50} = 4.16 \text{ cm}$.
2. Problem: Cylinder A has a radius $r$ and height $h$. Cylinder B has a radius $2r$ and height $0.5h$. What is the ratio of the volume of Cylinder A to Cylinder B?
   1. Write the formula for Cylinder A: $V_A = \pi r^2 h$.
   2. Write the formula for Cylinder B: $V_B = \pi (2r)^2 (0.5h)$.
   3. Simplify the expression for Cylinder B: $V_B = \pi (4r^2) (0.5h) = 2\pi r^2 h$.
   4. Compare the two: $\frac{V_A}{V_B} = \frac{\pi r^2 h}{2\pi r^2 h} = \frac{1}{2}$. The ratio is 1:2.
3. Problem: A cube has a total surface area of 150 square inches. What is its volume in cubic inches?
   1. A cube has 6 identical square faces. Set up the equation for surface area: $6s^2 = 150$.
   2. Divide by 6: $s^2 = 25$.
   3. Take the square root: $s = 5$.
   4. Calculate volume: $V = s^3 = 5^3 = 125 \text{ cubic inches}$.

Practice Questions

1. A rectangular box has dimensions 4, 6, and 10. If all dimensions are increased by 50%, by what percentage does the volume increase?

2. A cylindrical pipe is 20 feet long and has an inner radius of 3 inches. What is the internal volume of the pipe in cubic feet? (Note: 12 inches = 1 foot).

3. If the volume of a cube is 64, what is the total surface area of the cube?

4. Quantity A: The volume of a cylinder with radius 5 and height 10.
Quantity B: The volume of a rectangular solid with dimensions 8, 10, and 10.

5. A sphere is inscribed inside a cube with a side length of 6. What is the volume of the sphere in terms of $\pi$? (Formula for sphere volume: $V = \frac{4}{3}\pi r^3$).

6. Water is poured into a cylindrical container with a radius of 4 cm at a rate of $16\pi \text{ cm}^3$ per second. How many seconds will it take for the water level to rise by 10 cm?

7. A rectangular solid has a volume of 120. If the length is 5 and the width is 4, what is the length of the longest diagonal inside the solid?

8. If the radius of a cylinder is doubled and the height is tripled, the new volume is how many times the original volume?

9. A hollow metal cube has an outer side of 5 cm and an inner side of 4 cm. What is the volume of the metal used to make the cube?

10. A rectangular room has a volume of 1,200 cubic feet. If the floor area is 150 square feet, what is the height of the ceiling?

Answers & Explanations

1. Answer: 237.5%
   The original volume is $4 \times 6 \times 10 = 240$. If dimensions increase by 50%, they become 6, 9, and 15. The new volume is $6 \times 9 \times 15 = 810$. The increase is $810 - 240 = 570$. Percentage increase: $\frac{570}{240} \times 100 = 237.5\%$. (Shortcut: $1.5 \times 1.5 \times 1.5 = 3.375$, which is a 237.5% increase).
2. Answer: $1.25\pi$ or $\frac{5}{4}\pi$
   First, convert radius to feet: $3 \text{ inches} = 0.25 \text{ feet}$. Volume $V = \pi r^2 h = \pi (0.25)^2 (20) = \pi (0.0625)(20) = 1.25\pi \text{ cubic feet}$.
3. Answer: 96
   If $s^3 = 64$, then $s = 4$. Surface area $= 6s^2 = 6(4^2) = 6(16) = 96$.
4. Answer: Quantity B is greater
   Quantity A: $\pi (5^2)(10) = 250\pi \approx 250 \times 3.14 = 785$. Quantity B: $8 \times 10 \times 10 = 800$. Since 800 > 785, Quantity B is larger.
5. Answer: $36\pi$
   If the sphere is inscribed in a cube of side 6, its diameter is 6, so its radius $r = 3$. Volume $= \frac{4}{3}\pi (3^3) = \frac{4}{3}\pi (27) = 4\pi (9) = 36\pi$.
6. Answer: 10 seconds
   The volume needed to raise the level by 10 cm is $V = \pi (4^2)(10) = 160\pi \text{ cm}^3$. Time $= \frac{ \text{Volume}}{ \text{Rate}} = \frac{160\pi}{16\pi} = 10 \text{ seconds}$.
7. Answer: $\sqrt{77}$
   First find height: $5 \times 4 \times h = 120 \rightarrow 20h = 120 \rightarrow h = 6$. The diagonal of a rectangular solid is $\sqrt{l^2 + w^2 + h^2} = \sqrt{5^2 + 4^2 + 6^2} = \sqrt{25 + 16 + 36} = \sqrt{77}$.
8. Answer: 12
   Original: $V_1 = \pi r^2 h$. New: $V_2 = \pi (2r)^2 (3h) = \pi (4r^2)(3h) = 12\pi r^2 h$. The volume is 12 times larger.
9. Answer: 61
   Volume of outer cube $= 5^3 = 125$. Volume of inner empty space $= 4^3 = 64$. Metal volume $= 125 - 64 = 61 \text{ cm}^3$.
10. Answer: 8 feet
    Volume $= \text{Area of base} \times \text{height}$. $1,200 = 150 \times h$. $h = \frac{1,200}{150} = 8 \text{ feet}$.

1. If the side of a cube is tripled, by what factor does the volume increase?

• A3
• B9
• C27
• D81

Frequently Asked Questions

What is the difference between surface area and volume on the GRE?

Surface area measures the total area of all the exterior faces of a 3D object in square units, while volume measures the total space contained inside the object in cubic units. On the GRE, you may be asked to find one given the other, especially for cubes where both depend solely on the side length.

Do I need to memorize the volume of a sphere for the GRE?

While the GRE most commonly focuses on cubes, rectangular solids, and cylinders, the formula for the volume of a sphere ($V = \frac{4}{3}\pi r^3$) is occasionally provided in the question text. However, memorizing it can save time and increase your confidence during the exam.

How do I handle volume questions with different units?

Always convert all measurements to the same unit before performing calculations to avoid errors. For example, if the dimensions are in inches but the answer must be in cubic feet, it is usually easier to convert the inches to feet at the beginning of the problem. Check out Khan Academy's Geometry resources for more on unit conversion in 3D shapes.

What is a "right circular cylinder" on the GRE?

A right circular cylinder is a standard cylinder where the centers of the circular bases are aligned directly above one another, forming a 90-degree angle with the base. This is the only type of cylinder tested on the GRE, allowing you to use the standard $V = \pi r^2 h$ formula without worrying about slant heights.

Can I use the GRE calculator for volume problems?

Yes, the on-screen GRE calculator is available for volume calculations, which is particularly helpful when dealing with $\pi$ or large multiplications. However, many volume problems are designed so that $\pi$ cancels out or you can work with simple fractions, so look for algebraic simplifications first. To practice these mental shortcuts, you can use the Retrieval Challenge tool to sharpen your quick-recall of geometric properties.