Rectangular Prism Volume - SHSAT Math

A rectangular shipping box has interior dimensions of $4 \text{ in.}$ by $5 \text{ in.}$ by $6 \text{ in.}$ What is the volume of the box in cubic inches?
Answer: 120 When you encounter a question about the volume of a rectangular box, you're dealing with three-dimensional geometry. Volume measures how much space is inside a 3D shape, and for rectangular prisms (boxes), there's a straightforward formula to remember. To find the volume of any rectangular box, you multiply length × width × height. The interior dimensions given are 4 inches, 5 inches, and 6 inches. It doesn't matter which dimension you call length, width, or height - the multiplication will yield the same result. Volume = $4 \text{ in.} \times 5 \text{ in.} \times 6 \text{ in.} = 120 \text{ cubic inches}$ Now let's examine why the other answers are incorrect. Answer B (96) might result from incorrectly calculating $4 \times 6 \times 4 = 96$ - perhaps misreading one of the dimensions. Answer C (60) could come from multiplying only two dimensions: $4 \times 15 = 60$ or $5 \times 12 = 60$ , forgetting that volume requires all three dimensions. Answer D (15) represents the sum of the dimensions ( $4 + 5 + 6 = 15$ ) rather than their product - a fundamental conceptual error since volume involves multiplication, not addition. Remember this key distinction: perimeter and area problems often involve addition or multiplication of two dimensions, but volume always requires multiplying all three dimensions together. When you see "cubic" units in the answer choices, that's your signal that you need to find volume using length × width × height.
The figure shows a rectangular prism whose length, width, and height are three consecutive even integers. If the volume is $480$ cm³, what is the sum of the three dimensions, in centimeters?
Answer: $24$ Let the three consecutive even integers be n, n+2, and n+4. The volume is n(n+2)(n+4) = 480. Testing n = 6: 6 × 8 × 10 = 480 ✓. Therefore, the dimensions are 6 cm, 8 cm, and 10 cm. The sum is 6 + 8 + 10 = 24 cm.
A rectangular swimming pool is 25 meters long and 12 meters wide. If the pool holds 900 cubic meters of water when completely full, what is the depth of the pool?
Answer: 3.0 meters Using the volume formula for a rectangular prism: $V = l \times w \times h$ . We have $900 = 25 \times 12 \times h$ , so $900 = 300h$ , therefore $h = 3$ meters. Choice B is correct.
A rectangular tank is being filled with water at a rate of 3 cubic feet per minute. If the tank has a square base with sides of 6 feet and a height of 8 feet, how long will it take to fill the tank to 75% capacity?
Answer: 72 minutes First, find the total volume of the tank: $6 \times 6 \times 8 = 288$ cubic feet. For 75% capacity: $288 \times 0.75 = 216$ cubic feet. At a filling rate of 3 cubic feet per minute: $216 ÷ 3 = 72$ minutes. Choice A is correct.
A rectangular concrete foundation needs to be poured. The foundation is 27 feet long, 16 feet wide, and 2 feet deep. If concrete costs $120 per cubic yard, what is the total cost of the concrete needed?
Answer: $3,840 First find the volume in cubic feet: 27 × 16 × 2 = 864 cubic feet. Concrete is priced by the cubic yard. One cubic yard is 3 feet × 3 feet × 3 feet = 27 cubic feet, so divide by 27: 864 ÷ 27 = 32 cubic yards. Multiply by the price: 32 × $120 = $3,840. The most common mistake is dividing by 3 instead of 27, since 1 yard = 3 feet; that gives 288 cubic yards and a cost far too large. Another trap is dividing by 9, the number of square feet in a square yard, which gives 96 cubic yards. A cost of $4,320 would need 36 cubic yards, and $4,608 does not come from 32 cubic yards at $120. A cost of $1,440 would need only 12 cubic yards. The total cost of the concrete is $3,840.
A storage drawer measures $0.4\,\text{m}$ wide, $0.25\,\text{m}$ tall, and $0.60\,\text{m}$ deep. What is the volume of the drawer in cubic meters?
Answer: 0.060 When you encounter a problem asking for the volume of a rectangular object like a drawer, you need to apply the fundamental volume formula: Volume = length × width × height. The key is carefully multiplying all three dimensions and paying close attention to decimal placement. Here, you multiply the three given dimensions: $0.4 \times 0.25 \times 0.60$ . Let's work through this step by step. First, multiply $0.4 \times 0.25 = 0.1$ . Then multiply that result by the depth: $0.1 \times 0.60 = 0.060$ cubic meters. Looking at the wrong answers reveals common decimal errors. Choice B (0.0060) results from miscounting decimal places—when multiplying decimals, you should have exactly as many decimal places in your answer as the total decimal places in all factors combined. Here that's 1 + 2 + 2 = 5 decimal places, but 0.060 has effectively 3 significant decimal positions. Choice C (0.240) likely comes from adding the dimensions instead of multiplying them, a fundamental conceptual error. Choice D (6.00) suggests someone ignored the decimal points entirely and just multiplied 4 × 25 × 60, then converted incorrectly. The correct answer is A (0.060). Study tip: When working with decimal multiplication, count decimal places carefully and double-check your work by estimating. Here, $0.4 \times 0.25 \times 0.6$ is roughly $0.5 \times 0.25 \times 0.5 = 0.0625$ , which confirms that 0.060 is reasonable while eliminating the obviously incorrect choices.
In the figure, a rectangular prism has a base that is a $6$ in by $8$ in rectangle. The diagonal from one bottom corner to the opposite top corner has length $\sqrt{164}$ in. What is the volume of the prism, in cubic inches?
Answer: $384$ Space diagonal² = 6² + 8² + h² = 36 + 64 + h² = 100 + h² = 164, so h² = 64, h = 8. Volume = 6 × 8 × 8 = 384 in³. (A) uses h = 4. (B) uses h = 6. (D) uses h = 10.
In the figure, a rectangular prism has a volume of $720$ cm³. The area of the front face is $60$ cm² and the area of the top face is $80$ cm². What is the area of the side face (shaded), in square centimeters?
Answer: $108$ Let the dimensions be length l, width w, and height h. We know: lw = 80 (top face), lh = 60 (front face), and lwh = 720 (volume). From lwh = 720 and lw = 80, we get h = 720/80 = 9. From lh = 60 and h = 9, we get l = 60/9 = 20/3. From lw = 80 and l = 20/3, we get w = 80/(20/3) = 12. Therefore, the side face area is wh = 12 × 9 = 108 cm².
The rectangular fish tank shown below has interior dimensions of $30$ cm by $20$ cm by $25$ cm (height). Water is poured in until it reaches a height of $18$ cm. Then a solid rock is submerged in the tank, causing the water level to rise to $22$ cm (the water does not overflow). What is the volume of the rock, in cubic centimeters?
Answer: $2{,}400$ The rock displaces water equal to its volume. Water level rose by 22 − 18 = 4 cm. Displaced volume = 30 × 20 × 4 = 2,400 cm³. (A) uses rise of 4 with 20·20. (C) uses rise × incorrect base. (D) is the volume of water at height 22.
The rectangular tank shown in the figure is $40$ cm long, $30$ cm wide, and $50$ cm tall. It is tilted so that it rests on one of its $30$ cm edges, with water inside reaching exactly to the top edge on the lower side, forming a triangular prism of water. What is the volume of water in the tank, in cubic centimeters?
Answer: $30{,}000$ When tilted so water forms a triangular prism with the triangular cross section having legs 40 cm and 50 cm (along the length and height), the triangular area = ½(40)(50) = 1000 cm². Times the 30 cm width: V = 1000 × 30 = 30,000 cm³. (A) uses ½·40·25. (B) uses 40·30·20. (D) is the full tank volume·... — full tank is 60,000 cm³.
A rectangular wooden block is shown in the figure with dimensions $8$ cm × $6$ cm × $4$ cm. A cylindrical hole of radius $1$ cm is drilled through the block perpendicular to the $8$ cm × $6$ cm face, all the way through. Using $\pi \approx 3.14$ , what is the volume of the remaining solid, in cubic centimeters (to the nearest whole number)?
Answer: $179$ Volume of block = 8 × 6 × 4 = 192 cm³. The cylindrical hole has volume = π × r² × h = 3.14 × 1² × 4 = 12.56 cm³. Volume of remaining solid = 192 - 12.56 = 179.44 cm³, which rounds to 179 cm³.
A warehouse has dimensions 40 feet by 30 feet by 20 feet. If 60% of the warehouse space is occupied by stored goods, how many cubic feet of empty space remain?
Answer: 9,600 cubic feet Total warehouse volume: $40 \times 30 \times 20 = 24,000$ cubic feet. If 60% is occupied, then 40% is empty space. Empty space = $24,000 \times 0.40 = 9,600$ cubic feet. Choice B is correct.
Two rectangular prisms have the same volume of 360 cubic inches. Prism A has dimensions 6 inches by 8 inches by 7.5 inches. If Prism B has a length of 9 inches and a width of 5 inches, what is its height?
Answer: 8 inches When you encounter problems involving rectangular prisms with the same volume, you're working with the fundamental formula: Volume = length × width × height. The key insight is that different prisms can have the same volume despite having completely different dimensions. Since both prisms have a volume of 360 cubic inches, let's first verify Prism A: $6 × 8 × 7.5 = 360$ cubic inches. ✓ For Prism B, we know the length (9 inches), width (5 inches), and volume (360 cubic inches), but we need to find the height. Setting up the equation: $9 × 5 × h = 360$ Simplifying: $45h = 360$ Solving for h: $h = \frac{360}{45} = 8$ inches The answer is D) 8 inches. Let's examine why the other options are incorrect: A) 8.4 inches would give a volume of $9 × 5 × 8.4 = 378$ cubic inches, which exceeds the required 360. B) 6 inches would result in $9 × 5 × 6 = 270$ cubic inches, falling short of the target volume. C) 7.2 inches would produce $9 × 5 × 7.2 = 324$ cubic inches, also less than needed. Strategy tip: Always verify your volume calculations work both ways. After finding the missing dimension, multiply all three dimensions to confirm you get the stated volume. This catches arithmetic errors and ensures your answer makes sense within the context of the problem.
The rectangular prism shown in the figure has a square base. The perimeter of the base is $24$ cm, and the total surface area of the prism is $252$ cm². What is the volume of the prism, in cubic centimeters?
Answer: $270$ Each side of the square base is 24 ÷ 4 = 6 cm, so each base has area 6 × 6 = 36 cm². The surface area is two bases plus four rectangular sides: 2(36) + 4(6h) = 72 + 24h. Setting 72 + 24h = 252 gives 24h = 180, so h = 7.5 cm. Volume = base area × height = 36 × 7.5 = 270 cm³. 180 is the area of the four sides (252 − 72), not the volume. 216 is 6³, which assumes the prism is a cube with height 6 cm. 324 comes from counting only one base in the surface area: 36 + 24h = 252 gives h = 9, and 36 × 9 = 324.

Start this quiz