Volume & Surface Area - ACT Math

1. A cylinder has a height of 14 units and a radius of 3 units. What is the volume of the cylinder?
   Answer: 126π cubic units 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.
2. A rectangular prism has length 9 units, width 4 units, and height 2 units. What is the surface area of the rectangular prism?​
   Answer: 124 square units 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.
3. A cylinder has diameter 10 units and height 6 units. What is the volume of the cylinder? (Use $\pi$ in your answer.)
   Answer: $150\pi$ cubic units The solid is a cylinder with diameter 10 units (so radius r = 5 units) and height 6 units, and we are finding its volume. The formula for the volume of a cylinder is V = πr²h. Substituting r = 5 and h = 6, we get V = π × (5)² × 6 = π × 25 × 6. Then, 25 × 6 = 150, so V = 150π cubic units. Choice C, 600π cubic units, might result from using diameter instead of radius, as π(10)²×6 = π×100×6 = 600π.
4. A cone has radius 3 units and height 12 units. What is the volume?
   Answer: $36\pi$ cubic units 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.
5. A rectangular prism has dimensions 5 units by 4 units by 3 units. What is the surface area of the prism?
   Answer: 94 square units 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.
6. What is the surface area of a cube with side length 7 units?
   Answer: 294 square units This is finding the surface area of a cube with side length 7 units. The surface area formula for a cube is SA = 6s², where s is the side length, since a cube has six square faces. Substituting s = 7: SA = 6(7²) = 6(49) = 294 square units. Choice C uses s³ which gives volume, not surface area.
7. What is the surface area of a cube with side length 10 units?
   Answer: 600 square units 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.
8. A sphere has a volume of $288π$ cubic units. What is the radius of the sphere?
   Answer: 6 units This is finding the radius of a sphere given volume 288π cubic units. The volume formula for a sphere is V = (4/3)πr³. Setting up: 288π = (4/3)πr³, so 288 = (4/3)r³, which gives r³ = 288 × 3/4 = 216, therefore r = 6 units. We solve by isolating r³ first.
9. What is the volume of a rectangular prism with length 8 units, width 5 units, and height 3 units?
   Answer: 120 cubic units This problem asks for the volume of a rectangular prism with given dimensions. The volume formula for a rectangular prism is V = length × width × height. Substituting the given values: V = 8 × 5 × 3 = 40 × 3 = 120 cubic units. The other choices might result from adding dimensions instead of multiplying, or from calculating surface area instead of volume.
10. A rectangular prism has length 12 units, width 5 units, and height 3 units. All measurements are in units. What is the volume of the rectangular prism?
    Answer: 180 cubic units 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.
11. A cylinder has radius 4 units and height 10 units. What is the volume?
    Answer: $160\pi$ cubic units This problem asks for the volume of a cylinder with radius 4 units and height 10 units. The volume formula for a cylinder is V = πr²h, where r is the radius and h is the height. Substituting the given values: V = π(4²)(10) = π(16)(10) = 160π cubic units. Choice B might result from using 2πr instead of πr² in the formula.
12. A right circular cone has radius 3 units and height 12 units. What is the volume of the cone? (Use $\pi$ in your answer.)
    Answer: $36\pi$ cubic units The solid is a right circular cone with radius 3 units and height 12 units, and we are finding its volume. The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height. Substituting r = 3 and h = 12, we get V = (1/3)π × (3)² × 12 = (1/3)π × 9 × 12. First, 9 × 12 = 108, then (1/3) × 108 = 36, so V = 36π cubic units. Choice B, 108π cubic units, might result from forgetting the 1/3 factor in the formula.
13. What is the surface area of a sphere with radius 8 units?
    Answer: 256π square units We need to find the surface area of a sphere with radius 8 units. The surface area formula for a sphere is SA = 4πr² where r is the radius. Substituting r = 8: SA = 4π(8²) = 4π(64) = 256π square units. Choice B gives 128π, which is half the correct value, possibly from using 2πr² instead.
14. A square pyramid has a square base with side length 6 units and a vertical height of 9 units from the base to the apex. What is the volume of the pyramid in cubic units?
    Answer: 108 cubic units 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.
15. A cube has a surface area of 54 square units. What is the length of each side of the cube?
    Answer: 3 units We need to find the side length of a cube with surface area 54 square units. The surface area formula for a cube is SA = 6s² where s is the side length. Setting up the equation: 6s² = 54, so s² = 9, therefore s = 3 units. Each face of the cube has area s² = 9 square units.
16. A right circular cone has radius 3 units and height 12 units. The height is perpendicular to the circular base. What is the volume of the cone in cubic units?
    Answer: $36\pi$ cubic units The solid is a right circular cone with radius 3 units and height 12 units, and we need to find its volume. The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height. Substituting the values, V = (1/3)π(3)²(12) = (1/3)π(9)(12) = (1/3)π(108). This calculates to 36π cubic units. Omitting the 1/3 factor might lead to 108π.
17. A rectangular prism has dimensions 8 units, 6 units, and 5 units. What is the volume of the prism?
    Answer: 240 cubic units 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.
18. A cone has radius 3 units and height 12 units. What is the volume of the cone? (Use $\pi$ in your answer.)
    Answer: $36\pi\text{ cubic units}$ We need to find the volume of a cone with radius 3 units and height 12 units. The volume formula for a cone is V = (1/3)πr²h. Substituting r = 3 and h = 12: V = (1/3)π(3²)(12) = (1/3)π(9)(12) = (1/3)π(108) = 36π cubic units.
19. A cylinder has a surface area of 150π square units and a radius of 5 units. What is the height of the cylinder?
    Answer: 10 units We need to find the height of a cylinder with surface area 150π square units and radius 5 units. Using SA = 2πr² + 2πrh where r = 5: 150π = 2π(5²) + 2π(5)h = 50π + 10πh. Solving: 150π = 50π + 10πh, so 100π = 10πh, therefore h = 10 units.
20. A cylinder has diameter 12 units and height 5 units. What is the volume of the cylinder? (Use $\pi$ in your answer.)
    Answer: $180\pi\text{ cubic units}$ We need to find the volume of a cylinder with diameter 12 units and height 5 units. The volume formula for a cylinder is V = πr²h, where r is the radius. Since diameter = 12, radius r = 6. Substituting r = 6 and h = 5: V = π(6²)(5) = π(36)(5) = 180π cubic units.

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