Volume Formulas Test Answer Key with Scoring – Geometry

Question 1 Cylinder Volume (2 points)

V = π(4.5²)(8) = 162π ≈ 508.94.

≈ 508.9 cm³

Scoring: 1 point for the formula with the numbers, 1 point for the answer.

Question 2 Cone Volume (2 points)

V = 13π(6²)(11) = 132π ≈ 414.69.

≈ 414.7 in³

Scoring: 1 point for the one third, 1 point for the answer.

Question 3 Pyramid Volume (3 points)

Base area = 12 · 9 = 108. V = 13 · 108 · 10 = 360.

360 m³

Scoring: 1 point for the base area, 1 point for the one third, 1 point for the answer.

Question 4 Sphere Volume (2 points)

V = 43π(1.8³) = 43π(5.832) ≈ 24.43.

≈ 24.4 m³

Scoring: 1 point for the formula, 1 point for the answer.

Question 5 Cavalieri Explanation (2 points)

Both have the same height and equal cross-sectional area at every level (40 in²), so Cavalieri’s principle gives equal volumes. V = 40 · 9 = 360.

360 in³ each

Scoring: 1 point for naming equal slices and equal heights, 1 point for the volume.

Question 6 Capsule Volume (3 points)

Cylinder: π(3²)(10) = 90π. Two hemispheres make one sphere: 43π(3³) = 36π.

Total = 126π ≈ 395.84.

126π mm³ ≈ 395.8 mm³

Scoring: 1 point for the cylinder, 1 point for the sphere, 1 point for the total.

Question 7 Radius from Volume (3 points)

13πr²(9) = 75π, so 3r² = 75, r² = 25 and r = 5.

5 cm

Scoring: 1 point for the equation, 1 point for r², 1 point for the radius.

Question 8 Area of a Circle (3 points)

The base is half the circumference: ½ · 2π(6) = 6π cm. The height is the radius, 6 cm. Area ≈ 6π · 6 = 36π cm².

In general the base is πr and the height is r, so the area is πr · r = πr². More, thinner wedges make the shape closer to a true parallelogram.

Base 6π, height 6, area 36π cm²

Scoring: 1 point for base and height, 1 point for the area, 1 point for the explanation.