ISEE Middle Level Mathematics Achievement Quiz
Practice 3 D Volume in ISEE Middle Level Mathematics Achievement with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
Question 1 / 20
0 of 20 answered
A cubic box with an edge length of 10 inches is used to ship a smaller cubic box with an edge length of 6 inches. The remaining space in the larger box is filled with packing material. What is the volume of the packing material?
This quiz focuses on 3 D Volume, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Mathematics Achievement.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
A cubic box with an edge length of 10 inches is used to ship a smaller cubic box with an edge length of 6 inches. The remaining space in the larger box is filled with packing material. What is the volume of the packing material?
Explanation: First, find the volume of the larger box: V_large = 10³ = 10 × 10 × 10 = 1,000 cubic inches. Next, find the volume of the smaller box: V_small = 6³ = 6 × 6 × 6 = 216 cubic inches. The volume of the packing material is the difference between the two volumes: 1,000 - 216 = 784 cubic inches.
A large cube has an edge length that is three times the edge length of a small cube. The volume of the large cube is how many times the volume of the small cube?
Explanation: Let the edge length of the small cube be 's'. Its volume is s³. The edge length of the large cube is '3s'. Its volume is (3s)³ = 3s × 3s × 3s = 27s³. To find how many times larger the volume is, divide the large volume by the small volume: (27s³) / s³ = 27. The large cube's volume is 27 times that of the small cube.
A rectangular fish tank has a length of 30 cm and a width that is half its length. If the tank is filled with water to a height of 10 cm, what is the volume of the water in the tank?
Explanation: First, determine the dimensions of the water volume. The length is 30 cm. The width is half the length, so it is 30 cm / 2 = 15 cm. The height of the water is given as 10 cm. The volume of a rectangular prism is length × width × height. So, the volume of the water is 30 cm × 15 cm × 10 cm = 4,500 cubic cm.
A block of wood is a rectangular prism measuring 2 meters long, 50 centimeters wide, and 20 centimeters high. What is its volume in cubic centimeters?
Explanation: To calculate the volume in cubic centimeters, all dimensions must be in centimeters. The length is given as 2 meters. Since 1 meter = 100 centimeters, 2 meters = 200 centimeters. Now, multiply the dimensions: Volume = 200 cm × 50 cm × 20 cm = 10,000 × 20 = 200,000 cubic cm.
An L-shaped block is made from a large rectangular prism that is 10 m long, 4 m wide, and 3 m high, from which a smaller rectangular prism has been removed from one corner. The removed section was 6 m long, 2 m wide, and 3 m high. What is the volume of the L-shaped block?
Explanation: To find the volume of the L-shaped block, calculate the volume of the original large prism and subtract the volume of the removed section. Volume of the large prism = 10 m × 4 m × 3 m = 120 cubic m. Volume of the removed section = 6 m × 2 m × 3 m = 36 cubic m. The volume of the remaining block is 120 - 36 = 84 cubic m.
An object consists of a cone with a radius of 3 cm and a height of 10 cm, topped with a hemisphere with the same radius. What is the total volume of the object in terms of π? (Formulas: V_cone = (\frac{1}{3}\pi r^2h), V_sphere = (\frac{4}{3}\pi r^3))
Explanation: Calculate the volume of the cone and the hemisphere separately, then add them. V_cone = (\frac{1}{3}\pi (3^2)(10) = \frac{1}{3}\pi (9)(10) = 30\pi) cubic cm. A hemisphere is half a sphere. V_sphere = (\frac{4}{3}\pi (3^3) = \frac{4}{3}\pi (27) = 36\pi). V_hemisphere = (\frac{1}{2} \times 36\pi = 18\pi) cubic cm. Total volume = V_cone + V_hemisphere = 30π + 18π = 48π cubic cm.
A prism has a height of 12 cm. Its base is a right triangle with legs of 6 cm and 8 cm. What is the volume of the prism?
Explanation: The volume of a prism is the area of its base multiplied by its height. The base is a right triangle, so its area is (1/2) × base × height = (1/2) × 6 cm × 8 cm = 24 square cm. The height of the prism is 12 cm. Therefore, the volume of the prism is 24 cm² × 12 cm = 288 cubic cm.
A solid concrete block is a cube with a side length of 20 cm. A cylindrical hole with a diameter of 10 cm is drilled all the way through the center of one face to the opposite face. What is the volume of the concrete remaining in the block? Use 3 for π.
Explanation: When you encounter a problem involving removing material from a solid shape, you need to calculate the volume of the original shape minus the volume of the removed material. Start with the cube's volume: 203=8,000 cubic cm. Next, find the volume of the cylindrical hole. The diameter is 10 cm, so the radius is 5 cm. The cylinder goes through the entire cube, so its height equals the cube's side length (20 cm). Using the given value π = 3: Vcylinder=πr2h=3×52×20=3×25×20=1,500 cubic cm The remaining concrete volume is: 8,000−1,500=6,500 cubic cm, which is answer D. Looking at the wrong answers: A) 2,000 represents a major calculation error, possibly confusing dimensions or formulas entirely. B) 6,000 likely comes from incorrectly calculating the cylinder's volume as 2,000 instead of 1,500 - perhaps using the diameter (10) instead of radius (5) in the area calculation. C) 7,500 suggests subtracting only 500 from the cube's volume, which might result from miscalculating the cylinder's height or making an arithmetic error in the volume formula. Remember that "drilling through" means the hole goes completely through the object, so the cylinder's height equals the full dimension of the shape being drilled. Always double-check whether you're using radius or diameter in circular calculations - this is a frequent source of errors.
A square pyramid and a rectangular prism have bases with the same area. They also have the same height. If the volume of the prism is 36 cubic meters, what is the volume of the pyramid?
Explanation: The volume of a prism is V = Base Area × height. The volume of a pyramid is V = (1/3) × Base Area × height. Since the base area and height are the same for both shapes, the volume of the pyramid is exactly 1/3 of the volume of the prism. Therefore, the volume of the pyramid is (1/3) × 36 cubic meters = 12 cubic meters.
What is the volume of a sphere with a radius of 6 cm, in terms of π? Use the formula V = (\frac{4}{3}\pi r^3).
Explanation: Substitute the radius r = 6 cm into the volume formula for a sphere. V = (\frac{4}{3}\pi (6)^3). First, calculate 6³ = 6 × 6 × 6 = 216. Now, substitute this back into the formula: V = (\frac{4}{3}\pi (216)). To simplify, divide 216 by 3, which is 72. Then, multiply by 4: V = 4π(72) = 288π cubic cm.
Container A is a cylinder with a radius of 2 inches and a height of 9 inches. Container B is a cylinder with a radius of 3 inches and a height of 5 inches. Which statement accurately compares their volumes? Use π in your calculations.
Explanation: Calculate the volume of each container using V = πr²h. For Container A: V_A = π × (2²) × 9 = π × 4 × 9 = 36π cubic inches. For Container B: V_B = π × (3²) × 5 = π × 9 × 5 = 45π cubic inches. Comparing the two volumes, 45π is greater than 36π. Therefore, Container B has a greater volume.
A cube has a volume of 64 cubic inches. What is the total surface area of the cube?
Explanation: The formula for the volume of a cube is V = s³, where s is the side length. Given V = 64, we have s³ = 64. The cube root of 64 is 4, so the side length s = 4 inches. The formula for the surface area of a cube is SA = 6s². Substitute s = 4 into the formula: SA = 6 × (4²) = 6 × 16 = 96 square inches.
A cylindrical beaker with a radius of 5 cm is filled with water to a height of 15 cm. When a solid metal cube is completely submerged in the water, the water level rises to 17 cm. What is the volume of the metal cube? Use 3.14 for π.
Explanation: The volume of the submerged cube is equal to the volume of the water it displaces. The displaced water forms a cylindrical shape with the same radius as the beaker (5 cm) and a height equal to the change in water level. The water level rose from 15 cm to 17 cm, so the height of the displaced water is 17 - 15 = 2 cm. The volume of this displaced water is V = πr²h = 3.14 × (5²) × 2 = 3.14 × 25 × 2 = 157 cubic cm.
A pyramid has a square base with sides of 9 feet. The height of the pyramid is 10 feet. What is the volume of the pyramid?
Explanation: The formula for the volume of a pyramid is V = (1/3) × Base Area × height. The base is a square with sides of 9 feet, so the Base Area is 9 ft × 9 ft = 81 square feet. The height is 10 feet. Substitute these values into the formula: V = (1/3) × 81 sq ft × 10 ft = 27 × 10 = 270 cubic feet.
A rectangular swimming pool is 20 meters long, 10 meters wide, and 2 meters deep. It is being filled with water at a rate of 40 cubic meters per hour. How many hours will it take to fill the pool?
Explanation: First, calculate the total volume of the swimming pool: V = length × width × depth = 20 m × 10 m × 2 m = 400 cubic meters. To find the time it takes to fill the pool, divide the total volume by the fill rate: Time = Total Volume / Rate = 400 cubic meters / 40 cubic meters per hour = 10 hours.
A cylindrical storage tank has a volume of 785 cubic meters and a height of 10 meters. What is the radius of the tank's base? Use 3.14 for π.
Explanation: The formula for the volume of a cylinder is V = πr²h. We are given V = 785, h = 10, and π = 3.14. We need to solve for r. 785 = 3.14 × r² × 10. First, simplify the right side: 785 = 31.4 × r². Now, isolate r² by dividing both sides by 31.4: r² = 785 / 31.4 = 25. Finally, take the square root of both sides to find r: r = √25 = 5 meters.
A moving truck has a storage space that is 15 feet long, 8 feet wide, and 7 feet high. If boxes are cubes with an edge length of 2 feet, what is the maximum number of boxes that can fit in the truck, without being crushed or tilted?
Explanation: When you encounter a problem about fitting objects into a space, you're dealing with three-dimensional packing. The key is to think about how many objects can fit along each dimension separately, then multiply those numbers together. Let's work through each dimension of the truck. Along the 15-foot length, you can fit 15÷2=7.5 boxes, but since you can't use partial boxes, this means 7 complete boxes. Along the 8-foot width, you get 8÷2=4 boxes exactly. Along the 7-foot height, you can fit 7÷2=3.5 boxes, which means 3 complete boxes. The maximum number of boxes is 7×4×3=84 boxes, making D correct. Now let's see where the wrong answers come from. Choice A (105 boxes) likely results from incorrectly using the decimal values: 7.5×4×3.5=105. This ignores the fact that you can't fit partial boxes. Choice B (90 boxes) might come from miscalculating one dimension, perhaps thinking 6 boxes fit along the width instead of 4. Choice C (75 boxes) could result from various calculation errors, possibly confusing the dimensions or making arithmetic mistakes. Remember this strategy: always divide each dimension separately, round down to the nearest whole number (since partial objects don't count), then multiply. Don't be tempted to use decimal results in your final calculation—real objects must fit completely within the space.
A cylindrical can has a diameter of 10 feet and a height of 8 feet. What is the volume of the can? Use 3.14 for π.
Explanation: The formula for the volume of a cylinder is V = πr²h. The diameter is 10 feet, so the radius (r) is half of the diameter, which is 5 feet. The height (h) is 8 feet. Substitute the values into the formula: V = 3.14 × (5²) × 8 = 3.14 × 25 × 8 = 3.14 × 200 = 628 cubic feet.
The volume of a rectangular prism is 180 cubic inches. If the area of its base is 30 square inches, what is the height of the prism?
Explanation: The formula for the volume of a prism is Volume = Base Area × height. We are given the Volume (180 cubic inches) and the Base Area (30 square inches). To find the height, we can rearrange the formula: height = Volume / Base Area. So, height = 180 / 30 = 6 inches.
A jewelry box is in the shape of a rectangular prism. Its dimensions are (4\frac{1}{2}) inches by (3) inches by (2\frac{1}{3}) inches. What is the volume of the box in cubic inches?
Explanation: To find the volume, multiply the three dimensions. First, convert the mixed numbers to improper fractions: (4\frac{1}{2} = \frac{9}{2}) and (2\frac{1}{3} = \frac{7}{3}). Now, multiply the fractions: Volume = (\frac{9}{2} \times 3 \times \frac{7}{3}). The 3 in the numerator and the 3 in the denominator cancel out, leaving (\frac{9}{2} \times 7 = \frac{63}{2}). Convert this back to a mixed number: (63 \div 2 = 31) with a remainder of 1, so the volume is (31\frac{1}{2}) cubic inches.