This geometry volume formulas test answer key gives the full solution and the scoring guide for each of the 9 test questions on area and volume formulas.
Each solution shows the formula, the substitution and the result, followed by a scoring guide. Award partial credit only as the guide describes, and check that cubic units are used for every volume.
Geometry Volume Formulas Test Answer Key: questions 1–5
Question 1: Circle measures (2 points)
(a) \(C=\pi d=16\pi\approx 50.27\text{ in}\).
(b) \(r=8\), so \(A=\pi(8)^{2}=64\pi\approx 201.06\text{ in}^{2}\).
Answers: \(16\pi\text{ in}\); \(64\pi\text{ in}^{2}\).
Scoring: 1 point for the circumference, 1 point for the area (the radius must be 8, not 16).
Question 2: Rearranged sectors (2 points)
(a) The base is half the circumference: \(\frac{2\pi(12)}{2}=12\pi\text{ cm}\).
(b) The height is the radius, \(12\text{ cm}\). Area \(=12\pi\times 12=144\pi\text{ cm}^{2}\), which confirms \(A=\pi r^{2}\).
Answers: \(12\pi\text{ cm}\); \(144\pi\text{ cm}^{2}\).
Scoring: 1 point for the base; 1 point for the area and the formula.
Question 3: Slanted cylinder (2 points)
(a) The right cylinder has volume \(\pi(3)^{2}(10)=90\pi\text{ cm}^{3}\), and the slanted cylinder has the same volume.
(b) Cavalieri's principle: both solids are \(10\text{ cm}\) tall and every horizontal cross section is a circle of radius \(3\text{ cm}\) (area \(9\pi\text{ cm}^{2}\)), so the volumes are equal.
Answer: \(90\pi\text{ cm}^{3}\).
Scoring: 1 point for \(90\pi\); 1 point for naming Cavalieri's principle with equal heights and equal cross-sectional areas.
Question 4: Oil drum (2 points)
(a) \(r=12\), so \(V=\pi(12)^{2}(36)=5184\pi\text{ in}^{3}\) (about \(16{,}286.02\text{ in}^{3}\)).
(b) \(16{,}286.02: 231\approx 70.50\), so about \(70.5\) gallons.
Answers: \(5184\pi\text{ in}^{3}\); about \(70.5\) gallons.
Scoring: 1 point for using the radius 12 and getting \(5184\pi\); 1 point for the correct conversion to gallons.
Question 5: Pyramid height (2 points)
(a) \(150=\frac{1}{3}(36)h=12h\), so \(h=12.5\text{ m}\).
(b) The prism has \(V=Bh=36\times 12.5=450\text{ m}^{3}\), which is 3 times the pyramid.
Answers: \(12.5\text{ m}\); \(450\text{ m}^{3}\).
Scoring: 1 point for the height; 1 point for the prism volume.
Geometry Volume Formulas Test Answer Key: questions 6–9
Question 6: Pile of sand (2 points)
(a) \(r=5\), so \(V=\frac{1}{3}\pi(25)(6)=50\pi\approx 157.08\text{ ft}^{3}\).
(b) \(157.08\times 100=15{,}708\) pounds.
Answers: \(50\pi\approx 157.08\text{ ft}^{3}\); about \(15{,}708\) pounds.
Scoring: 1 point for the volume (radius 5, factor \(\frac{1}{3}\)); 1 point for the weight.
Question 7: Volume of a sphere (2 points)
\(V=\frac{4}{3}\pi(9)^{3}=\frac{4}{3}\pi(729)=972\pi\approx 3{,}053.63\).
Answer: \(972\pi\approx 3{,}053.63\text{ cm}^{3}\).
Scoring: 1 point for the correct formula and substitution; 1 point for \(972\pi\) and the rounded value.
Question 8: Hemisphere and cylinder (3 points)
(a) \(V=\pi(6)^{2}(6)=216\pi\text{ cm}^{3}\).
(b) \(V=\frac{1}{3}\pi(36)(6)=72\pi\text{ cm}^{3}\).
(c) \(216\pi-72\pi=144\pi\). The hemisphere is half a sphere: \(\frac{1}{2}\cdot\frac{4}{3}\pi(6)^{3}=\frac{2}{3}\pi(216)=144\pi\). The volumes are equal, as Cavalieri's principle predicts.
Answers: \(216\pi\); \(72\pi\); \(144\pi\text{ cm}^{3}\).
Scoring: 1 point for the cylinder; 1 point for the cone; 1 point for the subtraction and the matching hemisphere volume.
Question 9: Cone candles (3 points)
(a) \(\pi(4)^{2}(9)=144\pi\text{ cm}^{3}\).
(b) \(\frac{1}{3}\pi(3)^{2}(6)=18\pi\text{ cm}^{3}\).
(c) \(144\pi: 18\pi=8\) candles.
Answers: \(144\pi\text{ cm}^{3}\); \(18\pi\text{ cm}^{3}\); 8 candles.
Scoring: 1 point for each correct part.
Scoring summary
20 points: add the points of every question of this geometry volume formulas test answer key.
How to use this geometry volume formulas test answer key
Use this geometry volume formulas test answer key to score the test fairly and to see which skills need more practice.
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