Grade 8 Math Practice Test: Volume of Composite Solids and Missing Dimensions
- Volume of Composite Solids and Missing Dimensions
- 10 questions
- 15 minutes
This test uses V = πr²h for cylinders, V = 13πr²h for cones and V = 43πr³ for spheres, working backward to missing dimensions and combining solids. Tip: a hemisphere is half a sphere, and a cone holds one third of the cylinder with the same base and height.
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Question 1
A cylinder has a volume of 401.92 cm³ and a radius of 4 cm. What is its height? Use 3.14 for π.
Correct answer: C
The base area is 3.14 × 4² = 50.24 cm², and 401.92 ÷ 50.24 = 8 cm. 32 cm uses 3.14 × 4 instead of squaring the radius.
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Question 2
A cone has a volume of 96π cubic inches and a height of 8 inches. What is its radius?
Correct answer: C
13πr² · 8 = 96π, so r² = 96 × 3 ÷ 8 = 36 and r = 6 inches. 36 is r², not r.
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Question 3
A sphere has a volume of 36π cm³. What is its radius?
Correct answer: D
43πr³ = 36π, so r³ = 36 × 3 ÷ 4 = 27 and r = 3 cm. 27 is r³, and 6 cm is the diameter.
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Question 4
A grain silo is a cylinder with a radius of 3 m and a height of 10 m, topped by a hemisphere with the same radius. What is the total volume of the silo, in terms of π?
Correct answer: D
Cylinder: π · 3² · 10 = 90π. Hemisphere: half of 43π · 3³ = 18π. Total 108π m³. 126π adds a whole sphere (36π) instead of half of one.
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Question 5
A cone fits exactly inside a cylinder with the same base and the same height. The cylinder's volume is 150 cm³. What is the volume of the space inside the cylinder but outside the cone?
Correct answer: D
The cone holds one third of the cylinder: 150 ÷ 3 = 50 cm³. The space left over is 150 − 50 = 100 cm³. 50 cm³ is the cone itself.
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Question 6
A pipe is a hollow cylinder 20 cm long. Its outer radius is 5 cm and its inner radius is 4 cm. What is the volume of the material in the pipe, in terms of π?
Correct answer: A
Subtract the hole from the full cylinder: π · 5² · 20 − π · 4² · 20 = 500π − 320π = 180π cm³. 20π uses (5 − 4)² as if the wall were a single cylinder.
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Question 7
A cylindrical glass with a radius of 4 cm is filled with water to a height of 9 cm. The water is poured into an empty cylindrical vase with a radius of 6 cm. How deep is the water in the vase?
Correct answer: A
Water volume: π · 4² · 9 = 144π cm³. In the vase π · 6² · h = 144π, so h = 144 ÷ 36 = 4 cm. 6 cm scales the height by 4/6 instead of by (4/6)².
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Question 8
A cylindrical tub of ice cream has a radius of 6 cm and a height of 8 cm. Each scoop is a sphere with a radius of 2 cm. How many full scoops does the tub hold?
Correct answer: B
Tub: π · 6² · 8 = 288π cm³. Scoop: 43π · 2³ = 323π cm³. 288π ÷ 323π = 288 × 3 ÷ 32 = 27 scoops. 9 forgets the 43 and divides 288 by 32.
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Question 9
The radius of a cylinder is doubled and its height is cut in half. What happens to its volume?
Correct answer: B
New volume: π(2r)² · h2 = π · 4r² · h2 = 2πr²h, twice the original. "4 times as large" squares the doubled radius but forgets that the height was cut in half.
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Question 10
A medicine capsule is a cylinder 6 mm long with a radius of 2 mm, with a hemisphere of radius 2 mm on each end. What is its volume, to the nearest tenth? Use 3.14 for π.
Correct answer: A
Cylinder: 3.14 × 2² × 6 = 75.36 mm³. The two hemispheres make one sphere: 43 × 3.14 × 2³ ≈ 33.49 mm³. Total ≈ 108.9 mm³. 75.4 mm³ leaves out the rounded ends.
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