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Right Rectangular Prisms & Pyramids Unit Test
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Right Rectangular Prisms & Pyramids — Unit Test
Total: 100 points
Instructions: Show all work for short-answer and extended problems. Use π ≈ 3.14 only if asked. Unless otherwise stated, include units in answers.
Formulas you may use:
- Volume of a right rectangular prism: V = l × w × h
- Volume of a pyramid (any base): V = (1/3) × (area of base) × (height)
- Surface area of a rectangular prism: SA = 2(lw + lh + wh)
- Lateral area of a regular pyramid with n identical triangular faces: LA = (1/2) × (perimeter of base) × (slant height)
Part A — Multiple Choice (10 questions × 2 points = 20 points)
Circle the best answer.
1. (2 pts) A right rectangular prism has length 8 cm, width 3 cm, and height 5 cm. Its volume is:
A. 16 cm^3
B. 40 cm^3
C. 120 cm^3
D. 240 cm^3
2. (2 pts) A pyramid has base area 24 cm^2 and height 9 cm. Its volume is:
A. 72 cm^3
B. 216 cm^3
C. 648 cm^3
D. 8 cm^3
3. (2 pts) Which statement is always true?
A. A pyramid with the same base and same height as a prism has the same volume.
B. A pyramid with the same base and same height as a prism has 1/2 the volume of the prism.
C. A pyramid with the same base and same height as a prism has 1/3 the volume of the prism.
D. A pyramid with the same base and same height as a prism has 1/4 the volume of the prism.
4. (2 pts) Convert 12,000 cm^3 to m^3.
A. 1.2 × 10^-2 m^3
B. 0.012 m^3
C. 0.0012 m^3
D. 12 m^3
5. (2 pts) A cube has edge length 6 cm. If a pyramid has the same base (a 6 cm × 6 cm square) and same height as the cube, what is the pyramid’s volume relative to the cube’s volume?
A. 1/2 the cube’s volume
B. 1/3 the cube’s volume
C. 1/6 the cube’s volume
D. 1/9 the cube’s volume
6. (2 pts) A square pyramid with base side 4 cm has slant height 5 cm. The lateral surface area is:
A. 10 cm^2
B. 20 cm^2
C. 40 cm^2
D. 80 cm^2
7. (2 pts) If each linear dimension of a rectangular prism is doubled, the volume becomes:
A. doubled
B. tripled
C. quadrupled
D. eight times as large
8. (2 pts) A right rectangular prism has length 10 cm, width 6 cm, and height 2 cm. If the prism is filled to a height of 1.5 cm, the volume of water is:
A. 30 cm^3
B. 90 cm^3
C. 120 cm^3
D. 180 cm^3
9. (2 pts) The area of the base of a pyramid is 15 cm^2 and the pyramid’s volume is 30 cm^3. The height of the pyramid is:
A. 2 cm
B. 3 cm
C. 4 cm
D. 6 cm
10. (2 pts) A right rectangular prism has dimensions 4 cm by 5 cm by 6 cm. Its surface area is:
A. 94 cm^2
B. 148 cm^2
C. 180 cm^2
D. 300 cm^2
Part B — Short Answer, show all work (5 questions × 10 points = 50 points)
11. (10 pts) Find the volume of a rectangular prism with length 12 cm, width 7 cm, and height 3 cm. Include units.
12. (10 pts) A right square pyramid has base side 10 m and height 9 m. Find its volume. Include units and show work.
13. (10 pts) A rectangular prism measures 6 cm by 8 cm by 5 cm. On top of it sits a right rectangular pyramid whose base matches the prism’s top (6 × 8 cm) and whose height is 4 cm. Find the total volume of the combined solid (prism + pyramid). Include units.
14. (10 pts) The volume of a right rectangular prism is 360 cm^3. The base area (l × w) is 24 cm^2. Find the prism’s height. Show work.
15. (10 pts) A small rectangular prism measures 5 cm × 4 cm × 3 cm. A similar prism has linear scale factor 2/3 from the original (every dimension is multiplied by 2/3). Find the volume of the scaled prism. Show work.
Part C — Extended Problems (3 problems × 10 points = 30 points)
16. (10 pts) A rectangular water tank has inside dimensions length = 2.0 m, width = 1.5 m, height = 1.2 m. How many liters of water can it hold when full? (1 m^3 = 1000 L). Show work and include units.
17. (10 pts) A pyramid has rectangular base 10 cm by 6 cm. Its volume is 200 cm^3. Find the height of the pyramid. Show work and include units.
18. (10 pts) A solid consists of a rectangular prism of dimensions 30 cm (L) × 20 cm (W) × 15 cm (H). A square pyramid (base 20 cm by 20 cm) is removed (excavated) from the top center of the prism; the pyramid’s height is 5 cm and its base fits entirely on the top face of the prism. Find the volume of the remaining solid (prism minus pyramid). Show work.
Answer Key with Work and Scoring Notes
Part A — Answers (2 pts each)
1. C. 120 cm^3. (V = 8×3×5 = 120)
2. A. 72 cm^3. (V = (1/3)×24×9 = 72)
3. C. 1/3 the volume of the prism. (V_pyramid = (1/3)×B×h)
4. B. 0.012 m^3. (12,000 cm^3 × (1 m/100 cm)^3 = 12,000 × 1e-6 m^3 = 0.012 m^3)
5. B. 1/3 the cube’s volume. (Same base and height → pyramid is 1/3 of prism)
6. C. 40 cm^2. (Square pyramid: 4 triangles; lateral area = 4 × (1/2 × base × slant) = 4 × (1/2 × 4 × 5) = 4 × 10 = 40)
7. D. eight times as large. (Scale factor 2 → volume factor 2^3 = 8)
8. B. 90 cm^3. (Base area = 10×6 = 60; filled height 1.5 → volume = 60×1.5 = 90)
9. A. 2 cm. (V = (1/3) B h → 30 = (1/3)(15) h → 30 = 5h → h = 6? Wait—check: 30 = (1/3)(15)h → (1/3)*15 = 5 so h = 30/5 = 6. Correct option is D. 6 cm.)
Correction: 9: D. 6 cm.
10. A. 94 cm^2. (SA = 2(lw + lh + wh) = 2(4×5 + 4×6 + 5×6) = 2(20 + 24 + 30) = 2(74) = 148 — hold on check arithmetic.)
Let's compute carefully:
lw = 4×5 = 20
lh = 4×6 = 24
wh = 5×6 = 30
Sum = 20+24+30 = 74
SA = 2×74 = 148
So correct answer B. 148 cm^2.
Fix final answers for 9 and 10:
9. D. 6 cm.
10. B. 148 cm^2.
(Scoring note: each MC is 2 pts. For any arithmetic mistakes, deduct accordingly.)
Part B — Short answer (show work; full credit = 10 pts each)
11. Volume = l×w×h = 12×7×3 = 252 cm^3. (Answer: 252 cm^3)
12. Square base side = 10 m → base area B = 10×10 = 100 m^2. Volume = (1/3)×B×h = (1/3)×100×9 = 300 m^3. (Answer: 300 m^3)
13. Prism volume = 6×8×5 = 240 cm^3.
Pyramid base area B = 6×8 = 48 cm^2. Pyramid volume = (1/3)×48×4 = (1/3)×192 = 64 cm^3.
Total volume = 240 + 64 = 304 cm^3. (Answer: 304 cm^3)
14. V = base area × height → 360 = 24 × h → h = 360 / 24 = 15 cm. (Answer: 15 cm)
15. Original volume = 5×4×3 = 60 cm^3.
Scale factor for length = 2/3, so volume scale factor = (2/3)^3 = 8/27.
Scaled volume = 60 × 8/27 = (60/27)×8 = (20/9)×8 = 160/9 ≈ 17.777... cm^3.
Exact answer = 160/9 cm^3 ≈ 17.78 cm^3. (Show work and include fraction or decimal.)
Part C — Extended (full work shows reasoning)
16. Tank volume = 2.0×1.5×1.2 = 3.6 m^3.
Convert to liters: 3.6 m^3 × 1000 L/m^3 = 3600 L.
Answer: 3600 liters. (10 pts)
17. V = (1/3) × B × h. Base B = 10 × 6 = 60 cm^2.
Given V = 200 cm^3 → 200 = (1/3) × 60 × h → (1/3)×60 = 20 → 200 = 20h → h = 200/20 = 10 cm.
Answer: height = 10 cm. (10 pts)
18. Prism volume = 30 × 20 × 15 = 9000 cm^3.
Pyramid removed: base is 20 × 20 so B = 400 cm^2, height = 5 cm.
Pyramid volume = (1/3) × 400 × 5 = (1/3) × 2000 = 2000/3 ≈ 666.666... cm^3.
Remaining volume = 9000 − 2000/3 = (27,000/3 − 2000/3) = 25,000/3 ≈ 8333.333... cm^3.
Exact answer: 25,000/3 cm^3 ≈ 8333.33 cm^3.
(10 pts)
Scoring rubric suggestions:
- For Part B & C: award partial credit for correct intermediate steps. E.g., if student computes base area correctly but misapplies formula, give partial credit (about half).
- For unit mistakes: deduct 1–2 points if the numeric answer is correct but units missing/wrong.
- For Part A: each correct multiple-choice = 2 pts, no partial credit.
Would you like:
- a printable PDF or Word version of this test?
- an alternate version with diagrams/nets for visual questions?
- randomized versions for classroom use (multiple different but equivalent tests)?
Tell me which and I’ll prepare it.
Right Rectangular Prisms & Pyramids — Unit Test
Total: 100 points
Instructions: Show all work for short-answer and extended problems. Use π ≈ 3.14 only if asked. Unless otherwise stated, include units in answers.
Formulas you may use:
- Volume of a right rectangular prism: V = l × w × h
- Volume of a pyramid (any base): V = (1/3) × (area of base) × (height)
- Surface area of a rectangular prism: SA = 2(lw + lh + wh)
- Lateral area of a regular pyramid with n identical triangular faces: LA = (1/2) × (perimeter of base) × (slant height)
Part A — Multiple Choice (10 questions × 2 points = 20 points)
Circle the best answer.
1. (2 pts) A right rectangular prism has length 8 cm, width 3 cm, and height 5 cm. Its volume is:
A. 16 cm^3
B. 40 cm^3
C. 120 cm^3
D. 240 cm^3
2. (2 pts) A pyramid has base area 24 cm^2 and height 9 cm. Its volume is:
A. 72 cm^3
B. 216 cm^3
C. 648 cm^3
D. 8 cm^3
3. (2 pts) Which statement is always true?
A. A pyramid with the same base and same height as a prism has the same volume.
B. A pyramid with the same base and same height as a prism has 1/2 the volume of the prism.
C. A pyramid with the same base and same height as a prism has 1/3 the volume of the prism.
D. A pyramid with the same base and same height as a prism has 1/4 the volume of the prism.
4. (2 pts) Convert 12,000 cm^3 to m^3.
A. 1.2 × 10^-2 m^3
B. 0.012 m^3
C. 0.0012 m^3
D. 12 m^3
5. (2 pts) A cube has edge length 6 cm. If a pyramid has the same base (a 6 cm × 6 cm square) and same height as the cube, what is the pyramid’s volume relative to the cube’s volume?
A. 1/2 the cube’s volume
B. 1/3 the cube’s volume
C. 1/6 the cube’s volume
D. 1/9 the cube’s volume
6. (2 pts) A square pyramid with base side 4 cm has slant height 5 cm. The lateral surface area is:
A. 10 cm^2
B. 20 cm^2
C. 40 cm^2
D. 80 cm^2
7. (2 pts) If each linear dimension of a rectangular prism is doubled, the volume becomes:
A. doubled
B. tripled
C. quadrupled
D. eight times as large
8. (2 pts) A right rectangular prism has length 10 cm, width 6 cm, and height 2 cm. If the prism is filled to a height of 1.5 cm, the volume of water is:
A. 30 cm^3
B. 90 cm^3
C. 120 cm^3
D. 180 cm^3
9. (2 pts) The area of the base of a pyramid is 15 cm^2 and the pyramid’s volume is 30 cm^3. The height of the pyramid is:
A. 2 cm
B. 3 cm
C. 4 cm
D. 6 cm
10. (2 pts) A right rectangular prism has dimensions 4 cm by 5 cm by 6 cm. Its surface area is:
A. 94 cm^2
B. 148 cm^2
C. 180 cm^2
D. 300 cm^2
Part B — Short Answer, show all work (5 questions × 10 points = 50 points)
11. (10 pts) Find the volume of a rectangular prism with length 12 cm, width 7 cm, and height 3 cm. Include units.
12. (10 pts) A right square pyramid has base side 10 m and height 9 m. Find its volume. Include units and show work.
13. (10 pts) A rectangular prism measures 6 cm by 8 cm by 5 cm. On top of it sits a right rectangular pyramid whose base matches the prism’s top (6 × 8 cm) and whose height is 4 cm. Find the total volume of the combined solid (prism + pyramid). Include units.
14. (10 pts) The volume of a right rectangular prism is 360 cm^3. The base area (l × w) is 24 cm^2. Find the prism’s height. Show work.
15. (10 pts) A small rectangular prism measures 5 cm × 4 cm × 3 cm. A similar prism has linear scale factor 2/3 from the original (every dimension is multiplied by 2/3). Find the volume of the scaled prism. Show work.
Part C — Extended Problems (3 problems × 10 points = 30 points)
16. (10 pts) A rectangular water tank has inside dimensions length = 2.0 m, width = 1.5 m, height = 1.2 m. How many liters of water can it hold when full? (1 m^3 = 1000 L). Show work and include units.
17. (10 pts) A pyramid has rectangular base 10 cm by 6 cm. Its volume is 200 cm^3. Find the height of the pyramid. Show work and include units.
18. (10 pts) A solid consists of a rectangular prism of dimensions 30 cm (L) × 20 cm (W) × 15 cm (H). A square pyramid (base 20 cm by 20 cm) is removed (excavated) from the top center of the prism; the pyramid’s height is 5 cm and its base fits entirely on the top face of the prism. Find the volume of the remaining solid (prism minus pyramid). Show work.
Answer Key with Work and Scoring Notes
Part A — Answers (2 pts each)
1. C. 120 cm^3. (V = 8×3×5 = 120)
2. A. 72 cm^3. (V = (1/3)×24×9 = 72)
3. C. 1/3 the volume of the prism. (V_pyramid = (1/3)×B×h)
4. B. 0.012 m^3. (12,000 cm^3 × (1 m/100 cm)^3 = 12,000 × 1e-6 m^3 = 0.012 m^3)
5. B. 1/3 the cube’s volume. (Same base and height → pyramid is 1/3 of prism)
6. C. 40 cm^2. (Square pyramid: 4 triangles; lateral area = 4 × (1/2 × base × slant) = 4 × (1/2 × 4 × 5) = 4 × 10 = 40)
7. D. eight times as large. (Scale factor 2 → volume factor 2^3 = 8)
8. B. 90 cm^3. (Base area = 10×6 = 60; filled height 1.5 → volume = 60×1.5 = 90)
9. A. 2 cm. (V = (1/3) B h → 30 = (1/3)(15) h → 30 = 5h → h = 6? Wait—check: 30 = (1/3)(15)h → (1/3)*15 = 5 so h = 30/5 = 6. Correct option is D. 6 cm.)
Correction: 9: D. 6 cm.
10. A. 94 cm^2. (SA = 2(lw + lh + wh) = 2(4×5 + 4×6 + 5×6) = 2(20 + 24 + 30) = 2(74) = 148 — hold on check arithmetic.)
Let's compute carefully:
lw = 4×5 = 20
lh = 4×6 = 24
wh = 5×6 = 30
Sum = 20+24+30 = 74
SA = 2×74 = 148
So correct answer B. 148 cm^2.
Fix final answers for 9 and 10:
9. D. 6 cm.
10. B. 148 cm^2.
(Scoring note: each MC is 2 pts. For any arithmetic mistakes, deduct accordingly.)
Part B — Short answer (show work; full credit = 10 pts each)
11. Volume = l×w×h = 12×7×3 = 252 cm^3. (Answer: 252 cm^3)
12. Square base side = 10 m → base area B = 10×10 = 100 m^2. Volume = (1/3)×B×h = (1/3)×100×9 = 300 m^3. (Answer: 300 m^3)
13. Prism volume = 6×8×5 = 240 cm^3.
Pyramid base area B = 6×8 = 48 cm^2. Pyramid volume = (1/3)×48×4 = (1/3)×192 = 64 cm^3.
Total volume = 240 + 64 = 304 cm^3. (Answer: 304 cm^3)
14. V = base area × height → 360 = 24 × h → h = 360 / 24 = 15 cm. (Answer: 15 cm)
15. Original volume = 5×4×3 = 60 cm^3.
Scale factor for length = 2/3, so volume scale factor = (2/3)^3 = 8/27.
Scaled volume = 60 × 8/27 = (60/27)×8 = (20/9)×8 = 160/9 ≈ 17.777... cm^3.
Exact answer = 160/9 cm^3 ≈ 17.78 cm^3. (Show work and include fraction or decimal.)
Part C — Extended (full work shows reasoning)
16. Tank volume = 2.0×1.5×1.2 = 3.6 m^3.
Convert to liters: 3.6 m^3 × 1000 L/m^3 = 3600 L.
Answer: 3600 liters. (10 pts)
17. V = (1/3) × B × h. Base B = 10 × 6 = 60 cm^2.
Given V = 200 cm^3 → 200 = (1/3) × 60 × h → (1/3)×60 = 20 → 200 = 20h → h = 200/20 = 10 cm.
Answer: height = 10 cm. (10 pts)
18. Prism volume = 30 × 20 × 15 = 9000 cm^3.
Pyramid removed: base is 20 × 20 so B = 400 cm^2, height = 5 cm.
Pyramid volume = (1/3) × 400 × 5 = (1/3) × 2000 = 2000/3 ≈ 666.666... cm^3.
Remaining volume = 9000 − 2000/3 = (27,000/3 − 2000/3) = 25,000/3 ≈ 8333.333... cm^3.
Exact answer: 25,000/3 cm^3 ≈ 8333.33 cm^3.
(10 pts)
Scoring rubric suggestions:
- For Part B & C: award partial credit for correct intermediate steps. E.g., if student computes base area correctly but misapplies formula, give partial credit (about half).
- For unit mistakes: deduct 1–2 points if the numeric answer is correct but units missing/wrong.
- For Part A: each correct multiple-choice = 2 pts, no partial credit.
Would you like:
- a printable PDF or Word version of this test?
- an alternate version with diagrams/nets for visual questions?
- randomized versions for classroom use (multiple different but equivalent tests)?
Tell me which and I’ll prepare it.
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