Hard GRE Volume Questions Practice Questions

Calculate the capacity of a three-dimensional container by multiplying its base area by its height, a fundamental skill tested in Hard GRE Volume Questions. While basic geometry might focus on simple cubes, the GRE Quantitative Reasoning section often presents complex scenarios involving inscribed solids, rate-based filling problems, and proportional changes. Success on these high-level problems requires more than just memorizing formulas; you must be able to visualize spatial relationships and manipulate variables within the context of GRE Prep strategies.

Concept Explanation

Hard GRE Volume Questions typically involve three-dimensional shapes like rectangular prisms, cylinders, right circular cones, and spheres where the dimensions are not explicitly given or are altered by a specific percentage. Volume is the measure of how much space a 3D object occupies, and for the GRE, you must master the following standard formulas:

• Rectangular Prism: $V = l \times w \times h$
• Cylinder: $V = \pi r^2 h$
• Sphere: $V = \frac{4}{3} \pi r^3$
• Right Circular Cone: $V = \frac{1}{3} \pi r^2 h$

In harder variations, questions often combine these shapes or require you to find a volume after a shape has been submerged in a liquid (Archimedes' Principle). You might also encounter "optimization" style problems or questions where the ratio of volumes is compared when linear dimensions are scaled. A key rule to remember is that if the linear dimensions of a solid are multiplied by a factor of $k$, the volume is multiplied by a factor of $k^3$. This is a frequent shortcut in GRE practice questions with explanations that helps save time during the actual exam.

Solved Examples

1. Example 1: Inscribed Solids
   A sphere is perfectly inscribed inside a cube with a side length of 6 units. What is the volume of the space inside the cube that is NOT occupied by the sphere?
   
   1. Calculate the volume of the cube: $V_{ \text{cube}} = s^3 = 6^3 = 216$.
   2. Identify the radius of the sphere. Since it is inscribed, the diameter of the sphere equals the side of the cube (6), so the radius $r = 3$.
   3. Calculate the volume of the sphere: $V_{ \text{sphere}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi$.
   4. Subtract the sphere's volume from the cube's volume: $216 - 36\pi$.
2. Example 2: Rate and Time
   A cylindrical tank with a radius of 2 meters and a height of 5 meters is being filled with water at a rate of 3 cubic meters per minute. How many minutes will it take to fill the tank to 80% capacity? (Use $\pi \approx 3.14$)
   
   1. Find the total volume: $V = \pi (2^2)(5) = 20\pi \approx 62.8$ cubic meters.
   2. Find 80% of the volume: $0.80 \times 62.8 = 50.24$ cubic meters.
   3. Divide the target volume by the fill rate: $\frac{50.24}{3} \approx 16.75$ minutes.
3. Example 3: Scaling Dimensions
   If the radius of a right circular cylinder is doubled and its height is decreased by 50%, what is the ratio of the new volume to the original volume?
   
   1. Let the original volume be $V_1 = \pi r^2 h$.
   2. Define the new dimensions: $r_{ \text{new}} = 2r$ and $h_{ \text{new}} = 0.5h$.
   3. Calculate the new volume: $V_2 = \pi (2r)^2 (0.5h) = \pi (4r^2)(0.5h) = 2\pi r^2 h$.
   4. Determine the ratio: $V_2 : V_1 = 2\pi r^2 h : \pi r^2 h = 2:1$.

Practice Questions

1. A rectangular aquarium measures 10 inches by 12 inches by 15 inches. If it is currently half-full of water, and a solid metal cube with side length 4 inches is submerged, how many cubic inches of water would need to be added to fill the aquarium to the top?
2. A cone and a cylinder have the same height and the same base radius. If the volume of the cylinder is 120 cubic centimeters, what is the combined volume of the two shapes?
3. A spherical balloon is being inflated. If its surface area increases by 300%, by what percentage does its volume increase?

4. A solid metal sphere with a radius of 6 cm is melted down and recast into 8 identical smaller spheres. What is the radius of each smaller sphere?
5. A cylindrical pipe has an outer diameter of 10 cm, an inner diameter of 8 cm, and a length of 50 cm. What is the volume of the material used to make the pipe?
6. A right circular cylinder is inscribed in a cube such that the bases of the cylinder are tangent to the top and bottom faces of the cube. If the volume of the cube is 64, what is the volume of the cylinder?
7. A hemispherical bowl of radius 9 cm is full of liquid. This liquid is to be poured into cylindrical bottles of diameter 3 cm and height 4 cm. How many bottles are required to empty the bowl?
8. The volume of a cube is numerically equal to its surface area. What is the length of one edge of the cube?
9. If the height of a cone is increased by 20% and the radius of its base is decreased by 20%, what is the percentage change in its volume?
10. A rectangular tank with base dimensions 4m by 5m contains water to a depth of 2m. If a heavy stone of volume 10 cubic meters is dropped into the tank and is completely submerged, what is the new depth of the water?

Answers & Explanations

1. Answer: 836 cubic inches. The total volume is $10 \times 12 \times 15 = 1800$. Half-full means 900 cubic inches of space are empty. The cube takes up $4^3 = 64$ cubic inches of space. Therefore, the remaining empty space is $900 - 64 = 836$.
2. Answer: 160 cubic cm. The volume of a cylinder is $V = \pi r^2 h$ and a cone is $\frac{1}{3} \pi r^2 h$. If the cylinder is 120, the cone is $1/3 \times 120 = 40$. Total = $120 + 40 = 160$.
3. Answer: 700%. Surface area is proportional to $r^2$. An increase of 300% means the new area is 4 times the old (factor of 4). Thus, the radius increased by a factor of $\sqrt{4} = 2$. Volume is proportional to $r^3$, so the new volume is $2^3 = 8$ times the old. An 8-fold increase is a 700% increase.
4. Answer: 3 cm. Total volume $V = \frac{4}{3} \pi (6^3) = 288\pi$. Each small sphere has volume $288\pi / 8 = 36\pi$. Setting $\frac{4}{3} \pi r^3 = 36\pi$ gives $r^3 = 27$, so $r = 3$.
5. Answer: 450π cubic cm. Volume of material = Outer Volume - Inner Volume. Outer radius = 5, inner radius = 4. $V = \pi (5^2)(50) - \pi (4^2)(50) = 50\pi (25 - 16) = 50\pi (9) = 450\pi$.
6. Answer: 16π. Cube side $s = \sqrt[3]{64} = 4$. The cylinder's height is 4 and its diameter is 4 (radius = 2). $V = \pi (2^2)(4) = 16\pi$.
7. Answer: 54 bottles. Volume of hemisphere = $\frac{2}{3} \pi (9^3) = 486\pi$. Volume of one bottle (radius 1.5) = $\pi (1.5^2)(4) = 9\pi$. Number of bottles = $486\pi / 9\pi = 54$.
8. Answer: 6. Volume $s^3 = 6s^2$ (Surface Area). Dividing by $s^2$ gives $s = 6$.
9. Answer: 23.2% decrease. New volume = $\frac{1}{3} \pi (0.8r)^2 (1.2h) = \frac{1}{3} \pi (0.64r^2) (1.2h) = 0.768$ times the original. $1 - 0.768 = 0.232$ or 23.2% decrease.
10. Answer: 2.5 meters. The stone displaces 10 cubic meters of water. The base area is $4 \times 5 = 20$ sq meters. The rise in height $\text{h} = \text{Volume} / \text{Area} = 10 / 20 = 0.5$ meters. New depth = $2 + 0.5 = 2.5$.

1. If a cube's side length is tripled, by what factor does its volume increase?

• A3
• B9
• C27
• D81

Frequently Asked Questions

How often do volume questions appear on the GRE?

Geometry, including volume, typically makes up about 15% of the Quantitative Reasoning section. You can expect at least one or two 3D geometry questions per test, often at a high difficulty level.

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

Yes, the GRE does not provide a formula sheet. You must memorize the volumes of cylinders, rectangular solids, spheres, and cones to solve Hard GRE Volume Questions efficiently.

What is the most common trick in GRE volume problems?

The most common trick involves confusing radius with diameter or failing to convert units (e.g., mixing inches and feet). Always double-check the units and the specific dimension provided before calculating.

How do I handle volume questions with irregular shapes?

Most GRE shapes are standard. If you see an irregular shape, try to break it down into smaller, standard components like two rectangular prisms or a cylinder and a hemisphere.

Can I use a calculator for pi on the GRE?

The on-screen GRE calculator is basic. Most questions will either leave $\pi$ in the answer choices or expect you to use 3.14 or $22/7$ for an approximation.