ACT Math Quiz: Volume And Surface Area

What this quiz covers

This quiz focuses on Volume And Surface Area, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

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.

Question 1

1. 24
2. 120
3. 240 (correct answer)
4. 480

Explanation: This is a volume and displacement question testing Archimedes' principle (water displacement). Choice C (240) is correct — the volume of the submerged rock equals the volume of water displaced. The water level rose 2 inches across a 12 × 10 inch base. Volume displaced = 12 × 10 × 2 = 240 cubic inches = volume of rock. Choice A (24) adds the dimensions instead of multiplying: 12 + 10 + 2 = 24. Choice B (120) uses only the base area without the height rise: 12 × 10 = 120 — finding the base area but not the volume of displaced water. Choice D (480) doubles the correct answer — perhaps computing 12 × 10 × 4 (using 4 instead of 2) or multiplying the result by 2. Pro tip: When an object is submerged in a tank, the volume of the displaced water equals the volume of the object. Displaced water forms a rectangular prism with the tank's base dimensions and the height equal to the water rise. Volume = length × width × rise = 12 × 10 × 2 = 240. This principle applies whenever the tank has uniform (rectangular) cross-section.

Question 2

1. 72 square units
2. 124 square units (correct answer)
3. 152 square units
4. 104 square units

Explanation: We need to find the surface area of a rectangular prism with length 9, width 4, and height 2 units. The surface area formula is SA = 2(lw + lh + wh). Substituting: SA = 2(9×4 + 9×2 + 4×2) = 2(36 + 18 + 8) = 2(62) = 124 square units. Choice A incorrectly calculated only the sum of areas without doubling for opposite faces.

Question 3

1. 3 (correct answer)
2. 4.5
3. 6
4. 9

Explanation: This is a cylinder volume question testing inverse use of the volume formula. Choice A (3) is correct — V = πr²h → 72π = πr²(8) → divide both sides by π: 72 = 8r² → r² = 9 → r = 3 inches. Choice B (4.5) results from dividing 72 by 8 to get 9, then halving instead of taking the square root: 9/2 = 4.5. Choice C (6) may result from computing 72/8 = 9 and then computing √(9 × 4) = 6 — an incorrect extra step. Choice D (9) correctly solves r² = 9 but reports r² rather than r — forgetting to take the square root. Pro tip: When solving V = πr²h for r, cancel π first (it divides out cleanly), then divide by h to isolate r², and THEN take the square root. Write out each step to avoid stopping at r².

Question 4

1. 60 cubic units
2. 180 cubic units (correct answer)
3. 120 cubic units
4. 360 cubic units

Explanation: The solid is a rectangular prism with length 12 units, width 5 units, and height 3 units, and we need to find its volume. The formula for the volume of a rectangular prism is V = lwh, where l is length, w is width, and h is height. Substituting the values, V = 12·5·3 = 60·3. This calculates to 180 cubic units. Confusing with surface area might lead to other values like 2(lw + lh + wh) = 222.

Question 5

1. $108\pi$ cubic units
2. $36\pi$ cubic units (correct answer)
3. $45\pi$ cubic units
4. 108 cubic units

Explanation: This problem asks for the volume of a cone with radius 3 units and height 12 units. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. Substituting the given values: V = (1/3)π(3²)(12) = (1/3)π(9)(12) = (1/3)π(108) = 36π cubic units. Choice A might result from forgetting the 1/3 factor in the cone volume formula.

Question 6

1. $36\pi\text{ cubic units}$ (correct answer)
2. $27\pi\text{ cubic units}$
3. $12\pi\text{ cubic units}$
4. $\frac{4}{3}\pi\text{ cubic units}$

Explanation: We need to find the volume of a sphere with radius 3 units. The volume formula for a sphere is V = (4/3)πr³. Substituting r = 3: V = (4/3)π(3³) = (4/3)π(27) = (4 × 27/3)π = (108/3)π = 36π cubic units.

Question 7

1. 72 cubic units (correct answer)
2. 52 cubic units
3. 84 cubic units
4. 96 cubic units

Explanation: This is finding the volume of a rectangular prism with length 6, width 4, and height 3 units. The volume formula for a rectangular prism is V = lwh, where l is length, w is width, and h is height. Substituting the values: V = 6 × 4 × 3 = 72 cubic units. This involves multiplying all three dimensions together.

Question 8

1. 108 cubic units (correct answer)
2. 324 cubic units
3. 216 cubic units
4. 72 cubic units

Explanation: The solid is a square pyramid with base side length 6 units and height 9 units, and we need to find its volume. The formula for the volume of a pyramid is V = (1/3)Bh, where B is the base area and h is the height. The base area B = 6² = 36, so V = (1/3)(36)(9) = (1/3)(324). This calculates to 108 cubic units. Forgetting the 1/3 factor might yield 324.

Question 9

1. 240 cubic units (correct answer)
2. 96 cubic units
3. 280 cubic units
4. 120 cubic units

Explanation: We need to find the volume of a rectangular prism with dimensions 8, 6, and 5 units. The volume formula for a rectangular prism is V = lwh where l, w, and h are the dimensions. Substituting: V = 8 × 6 × 5 = 240 cubic units. This represents the space occupied by the three-dimensional object.

Question 10

1. 300 square units
2. 1000 square units
3. 600 square units (correct answer)
4. 400 cubic units

Explanation: We need to find the surface area of a cube with side length 10 units. The surface area formula for a cube is SA = 6s² where s is the side length. Substituting s = 10: SA = 6(10²) = 6(100) = 600 square units. A cube has six square faces, each with area s².

Question 11

1. 94 square units (correct answer)
2. 60 square units
3. 72 square units
4. 120 square units

Explanation: We need to find the surface area of a rectangular prism with dimensions 5, 4, and 3 units. The surface area formula is SA = 2(lw + lh + wh) where l, w, h are the dimensions. Substituting: SA = 2(5×4 + 5×3 + 4×3) = 2(20 + 15 + 12) = 2(47) = 94 square units. This accounts for all six rectangular faces of the prism.

Question 12

1. 120 cubic units (correct answer)
2. 360 cubic units
3. 180 cubic units
4. 90 cubic units

Explanation: We need to find the volume of a rectangular pyramid with base area 40 square units and height 9 units. The volume formula for a pyramid is $V = \frac{1}{3} B h$ where B is the base area and h is the height. Substituting B = 40 and h = 9: $V = \frac{1}{3} \times 40 \times 9 = \frac{1}{3} \times 360 = 120$ cubic units. Choice B incorrectly omits the 1/3 factor.

Question 13

1. $2πr^2h$
2. $πr^2h$
3. $πr^2 + 2πrh$
4. $2πr^2 + 2πrh$ (correct answer)

Explanation: We need to identify the formula for the surface area of a cylinder with radius r and height h. The surface area consists of two circular bases (each with area πr²) and the curved lateral surface (with area 2πrh). Therefore, the total surface area is SA = 2πr² + 2πrh. Choice B gives only the volume formula πr²h.

Question 14

1. 81π cubic units
2. 27π cubic units (correct answer)
3. 54π cubic units
4. 36π cubic units

Explanation: We need to find the volume of a cone with radius 3 units and height 9 units. The volume formula for a cone is V = (1/3)πr²h where r is the radius and h is the height. Substituting r = 3 and h = 9: V = (1/3)π(3²)(9) = (1/3)π(9)(9) = (1/3)π(81) = 27π cubic units. Choice A incorrectly omits the 1/3 factor in the cone volume formula.

Question 15

1. 160π cubic units (correct answer)
2. 320π cubic units
3. 80π square units
4. 64π cubic units

Explanation: We need to find the volume of a cylinder with diameter 8 units and height 10 units. The volume formula for a cylinder is V = πr²h where r is the radius. Since diameter = 8, radius = 4 units. Substituting r = 4 and h = 10: V = π(4²)(10) = π(16)(10) = 160π cubic units. Choice D incorrectly uses the diameter instead of radius in the calculation.

Question 16

1. 126π cubic units (correct answer)
2. 84π cubic units
3. 252π cubic units
4. 168π cubic units

Explanation: We need to find the volume of a cylinder with height 14 units and radius 3 units. The volume formula for a cylinder is V = πr²h where r is the radius and h is the height. Substituting r = 3 and h = 14: V = π(3²)(14) = π(9)(14) = 126π cubic units. This represents the space inside the cylindrical container.

Question 17

1. 8 units
2. 10 units
3. 5 units (correct answer)
4. 15 units

Explanation: We need to find the radius of a cylinder with volume 300π cubic units and height 12 units. The volume formula for a cylinder is V = πr²h where r is the radius and h is the height. Setting up: πr²(12) = 300π, so 12r² = 300, therefore r² = 25, and r = 5 units.

Question 18

1. 75 square units
2. 25 square units
3. 150 cubic units
4. 125 cubic units (correct answer)

Explanation: We need to find the volume of a cube with side length 5 units. The volume formula for a cube is V = s³ where s is the side length. Substituting s = 5: V = 5³ = 5 × 5 × 5 = 125 cubic units. Choice B gives 25 square units, which incorrectly uses area units instead of volume units.

Question 19

1. 48 square units
2. 64 square units
3. 128 square units
4. 96 square units (correct answer)

Explanation: We need to find the surface area of a cube with volume 64 cubic units. First, find the side length using V = s³: s³ = 64, so s = 4 units. Then use the surface area formula SA = 6s² where s is the side length. Substituting s = 4: SA = 6(4²) = 6(16) = 96 square units. The cube has six square faces, each with area 16 square units.

Question 20

1. 3 units (correct answer)
2. 4 units
3. 5 units
4. 6 units

Explanation: This is finding the side length of a cube given surface area 54 square units. The surface area formula for a cube is SA = 6s², where s is the side length. Setting up: 54 = 6s², so s² = 54/6 = 9, therefore s = 3 units. The cube has six equal square faces.