Volume Formulas Worksheet Answer Key – Geometry

Problem 1 Area of a Circle from Wedges

(a) The base is half the circumference: ½ · 2π(8) = 8π. The height is the radius, 8.

(b) Area = base · height = 8π · 8 = 64π.

(a) base 8π in, height 8 in (b) 64π in²

Problem 2 Volume of a Cylinder

V = πr²h = π(5²)(12) = 300π.

300π ≈ 942.5.

300π in³ ≈ 942.5 in³

Problem 3 Cylinder from a Diameter

The radius is 14 ÷ 2 = 7. V = π(7²)(9) = 441π ≈ 1385.44.

≈ 1385.4 cm³

Problem 4 Volume of a Cone

V = 13π(6²)(10) = 13π · 360 = 120π.

120π ≈ 376.99.

120π m³ ≈ 377.0 m³

Problem 5 Volume of a Square Pyramid

Base area = 9 · 9 = 81. V = 13 · 81 · 14 = 27 · 14 = 378.

378 m³

Problem 6 Volume of a Sphere

V = 43π(9³) = 43π · 729 = 972π.

972π ≈ 3053.63.

972π cm³ ≈ 3053.6 cm³

Problem 7 Cavalieri and an Oblique Cylinder

(a) Both have the same height, and every horizontal slice of each is a circle of radius 4, so every pair of slices has equal area. By Cavalieri’s principle the volumes are equal. The slant side does not matter.

(b) V = π(4²)(10) = 160π ≈ 502.7.

160π ft³ ≈ 502.7 ft³

Problem 8 One Third of a Cube

(a) 6³ = 216.

(b) 216 ÷ 3 = 72. Formula: 13 · 36 · 6 = 72.

(a) 216 in³ (b) 72 in³

Problem 9 Height of a Cone

13π(6²)h = 96π, so 12h = 96 and h = 8.

8 in

Problem 10 Radius of a Cylinder

πr²(6) = 486π, so r² = 81 and r = 9.

9 m

Problem 11 Radius of a Sphere

43πr³ = 288π, so r³ = 288 · 34 = 216 and r = 6.

6 ft

Problem 12 Hemisphere Bowl

Hemisphere volume = ½ · 43π(10³) = 2000π3 ≈ 2094.4 cm³.

Liters: 2094.4 ÷ 1,000 ≈ 2.09.

≈ 2094.4 cm³ ≈ 2.1 L

Problem 13 Rectangular Pyramid

Base area = 8 · 15 = 120. V = 13 · 120 · 11 = 40 · 11 = 440.

440 ft³

Problem 14 Cone with a Slant Height

The height, radius and slant height form a right triangle: h = √(17² − 8²) = √225 = 15.

V = 13π(8²)(15) = 320π ≈ 1005.31.

h = 15 cm; V = 320π cm³ ≈ 1005.3 cm³

Problem 15 Grain Silo

Cylinder: π(6²)(24) = 864π. Cone: 13π(6²)(5) = 60π.

Total = 924π ≈ 2902.83.

924π ft³ ≈ 2903 ft³

Problem 16 Ice Cream Cone

Cone: 13π(3²)(11) = 33π. Hemisphere: 23π(3³) = 18π.

Total = 51π ≈ 160.22.

51π cm³ ≈ 160.2 cm³

Problem 17 Water Tank

Capacity = π(3²)(4.5) = 40.5π ≈ 127.23 m³.

80%: 0.8 · 127.23 ≈ 101.79 m³ = 101,788 L.

≈ 102,000 L

Problem 18 Melting a Sphere

Sphere volume = 43π(6³) = 288π.

Cylinder: π(4²)h = 16πh = 288π, so h = 18.

18 cm

Problem 19 Pyramid Monument

Monument: 13 · 32² · 21 = 13 · 1,024 · 21 = 7,168 m³.

Each dimension is divided by 8, so the volume is divided by 8³ = 512: 7,168 ÷ 512 = 14.

Monument 7,168 m³; model 14 m³

Problem 20 Sand Timer

Cone volume = 13π(3²)(6) = 18π.

Cylinder: π(3²)h = 9πh = 18π, so h = 2.

(Equal bases mean the cone fills one third of a cylinder of its own height, 6 ÷ 3 = 2.)

2 cm