Geometry Practice Test: Volume and Surface Area of Solids
- Volume and Surface Area
- 10 questions
- 15 minutes
Answers are given in terms of π where a circle is involved. Tip: check whether a question gives a radius or a diameter, and whether a cone or pyramid height is the slant height.
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Question 1
A cylinder has a radius of 4 cm and a height of 9 cm.
What is its total surface area?
Correct answer: B
Total surface area = 2πr² + 2πrh = 32π + 72π = 104π cm². The value 88π counts only one circular base.
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Question 2
A cone has a base radius of 5 in. and a height of 12 in.
What is its lateral (curved) surface area?
Correct answer: C
The slant height is 25 + 144 = 13 in., and the lateral area is πrℓ = π · 5 · 13 = 65π in². Using the height 12 instead of the slant height gives 60π, and adding the base gives 90π.
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Question 3
What is the surface area of a sphere with a radius of 6 m?
Correct answer: A
Surface area = 4πr² = 4π · 36 = 144π m². The value 288π is the volume of the same sphere.
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Question 4
What is the volume of a sphere with a diameter of 18 cm?
Correct answer: D
The radius is 9 cm, so V = (4/3)π · 9³ = (4/3)π · 729 = 972π cm³. Using 18 as the radius gives 7776π.
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Question 5
A prism has a right-triangle base with legs of 6 cm and 8 cm. The prism is 10 cm long.
What is its volume?
Correct answer: B
The base area is ½ · 6 · 8 = 24 cm², and the volume is 24 · 10 = 240 cm³. Forgetting the ½ for the triangle gives 480.
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Question 6
A square pyramid has a base edge of 10 ft and a slant height of 13 ft.
What is its total surface area?
Correct answer: D
The base is 10² = 100 ft², and each of the four triangular faces is ½ · 10 · 13 = 65 ft². The total is 100 + 260 = 360 ft². The value 260 is only the lateral area.
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Question 7
A cylindrical tank has a radius of 3 ft and a height of 7 ft.
To the nearest tenth, what is its volume?
Correct answer: C
V = πr²h = π · 9 · 7 = 63π ≈ 197.9 ft³. The value 66.0 is one third of that (the cone formula), and 791.7 uses the diameter 6 as the radius.
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Question 8
Two similar solids have a scale factor of 2 : 3. The smaller solid has a volume of 40 cm³.
What is the volume of the larger solid?
Correct answer: D
Volumes scale by the cube of the scale factor: 40 · (3/2)³ = 40 · 27/8 = 135 cm³. Using the square of the factor gives 90.
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Question 9
A cube has a total surface area of 150 cm².
What is its volume?
Correct answer: A
Each of the six faces is 150/6 = 25 cm², so an edge is 5 cm and the volume is 5³ = 125 cm³. The value 25 is the area of one face.
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Question 10
A solid is made of a cylinder with radius 3 cm and height 10 cm, with a cone of the same radius and height 4 cm on top.
What is the volume of the solid?
Correct answer: B
The cylinder holds π · 9 · 10 = 90π and the cone holds (1/3)π · 9 · 4 = 12π, for a total of 102π cm³. Treating the cone like a cylinder (36π) gives 126π.
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