This worksheet provides practice in calculating the volume of complex 3D shapes by adding or subtracting standard geometric solids.

I will solve each of the six problems for finding the volume of compound figures. I'll calculate the volume for each shaded region by breaking down the compound shapes into their basic geometric components, calculating the volume of each part, and then adding or subtracting as necessary.

Let's start with problem 1.

Problem 1

The figure is a rectangular prism with a pyramid on top.
- Rectangular prism: length = 20 ft, width = 18 ft, height = 4 ft
- Pyramid: base is the same as the top of the prism (20 ft by 18 ft), height = 7 ft

Volume of rectangular prism = length × width × height = 20 × 18 × 4 = 1440 ft³

Volume of pyramid = (1/3) × base area × height = (1/3) × (20 × 18) × 7 = (1/3) × 360 × 7 = 840 ft³

Total volume = 1440 + 840 = 2280.00 ft³

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Problem 2

This appears to be a triangular prism.
- Base of triangle: 26 in
- Height of triangle: 18 in
- Length of prism: 25 in

Volume of triangular prism = (1/2) × base × height × length = (1/2) × 26 × 18 × 25

First, (1/2) × 26 = 13
Then, 13 × 18 = 234
Finally, 234 × 25 = 5850 in³

Total volume = 5850.00 in³

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Problem 3

This is a rectangular prism with a cylindrical hole through it.
- Rectangular prism: length = 20 yd, width = 12 yd, height = 15 yd
- Cylinder: diameter = 5 yd (so radius = 2.5 yd), height = 12 yd (same as the width of the prism)

Volume of rectangular prism = 20 × 12 × 15 = 3600 yd³

Volume of cylinder = π × r² × h = 3.14 × (2.5)² × 12 = 3.14 × 6.25 × 12

First, 6.25 × 12 = 75
Then, 3.14 × 75 = 235.5 yd³

Volume of shaded region = Volume of prism - Volume of cylinder = 3600 - 235.5 = 3364.50 yd³

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Problem 4

This is a cylinder with a hemisphere on top.
- Cylinder: diameter = 16 ft (so radius = 8 ft), height = 9 ft
- Hemisphere: same radius = 8 ft

Volume of cylinder = π × r² × h = 3.14 × 8² × 9 = 3.14 × 64 × 9

First, 64 × 9 = 576
Then, 3.14 × 576 = 1808.64 ft³

Volume of hemisphere = (1/2) × (4/3) × π × r³ = (2/3) × π × r³ = (2/3) × 3.14 × 8³

First, 8³ = 512
Then, (2/3) × 3.14 × 512 = (2/3) × 1607.68 = 1071.7867... ft³

Total volume = 1808.64 + 1071.79 = 2880.43 ft³ (rounded to two decimal places)

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Problem 5

This is three spheres stacked on top of each other.
- Each sphere has a radius of 4 in.

Volume of one sphere = (4/3) × π × r³ = (4/3) × 3.14 × 4³ = (4/3) × 3.14 × 64

First, 3.14 × 64 = 200.96
Then, (4/3) × 200.96 = 267.9467... in³

Since there are three identical spheres:
Total volume = 3 × 267.9467... = 803.84 in³ (rounded to two decimal places)

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Problem 6

This is a rectangular prism with a square pyramid cut out from the top.
- Rectangular prism: length = 28 yd, width = 13 yd, height = 10 yd
- Square pyramid: base = 28 yd by 13 yd, height = 12 yd

Volume of rectangular prism = 28 × 13 × 10 = 3640 yd³

Volume of pyramid = (1/3) × base area × height = (1/3) × (28 × 13) × 12 = (1/3) × 364 × 12 = 1456 yd³

Volume of shaded region = Volume of prism - Volume of pyramid = 3640 - 1456 = 2184.00 yd³

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Here are the final answers for each problem:

1) 2280.00 ft³
2) 5850.00 in³
3) 3364.50 yd³
4) 2880.43 ft³
5) 803.84 in³
6) 2184.00 yd³