Volume of Composite Figures activity - Free Printable

To solve the problem of finding the volumes of the compound shapes, we need to break each shape into simpler geometric components and calculate their volumes individually. Then, we sum or subtract the volumes as necessary. Let's go through each problem step by step.

---

Problem 1:

The shape consists of a rectangular prism with a triangular pyramid on top.

#### Step 1: Volume of the Rectangular Prism
The dimensions of the rectangular prism are:
- Length = 16 ft
- Width = 8 ft
- Height = 8 ft

The formula for the volume of a rectangular prism is:
\[
V_{\text{prism}} = \text{Length} \times \text{Width} \times \text{Height}
\]
\[
V_{\text{prism}} = 16 \times 8 \times 8 = 1024 \, \text{ft}^3
\]

#### Step 2: Volume of the Triangular Pyramid
The triangular pyramid has:
- Base (triangle) dimensions: Base = 12 ft, Height = 8 ft
- Height of the pyramid = 8 ft

The area of the triangular base is:
\[
\text{Area of base} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 12 \times 8 = 48 \, \text{ft}^2
\]

The formula for the volume of a pyramid is:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
\[
V_{\text{pyramid}} = \frac{1}{3} \times 48 \times 8 = \frac{1}{3} \times 384 = 128 \, \text{ft}^3
\]

#### Step 3: Total Volume
The total volume is the sum of the volumes of the prism and the pyramid:
\[
V_{\text{total}} = V_{\text{prism}} + V_{\text{pyramid}} = 1024 + 128 = 1152 \, \text{ft}^3
\]

Answer for Problem 1:
\[
\boxed{1152}
\]

---

Problem 2:

The shape consists of a cylinder with a hemisphere on top.

#### Step 1: Volume of the Cylinder
The dimensions of the cylinder are:
- Radius = 9 cm
- Height = 12 cm

The formula for the volume of a cylinder is:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
\[
V_{\text{cylinder}} = 3.14 \times 9^2 \times 12 = 3.14 \times 81 \times 12 = 3.14 \times 972 = 3052.08 \, \text{cm}^3
\]

#### Step 2: Volume of the Hemisphere
The radius of the hemisphere is the same as the cylinder's radius, which is 9 cm.

The formula for the volume of a sphere is:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Since we have a hemisphere, its volume is half of the sphere's volume:
\[
V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 9^3 = \frac{2}{3} \times 3.14 \times 729 = \frac{2}{3} \times 2288.58 = 1525.72 \, \text{cm}^3
\]

#### Step 3: Total Volume
The total volume is the sum of the volumes of the cylinder and the hemisphere:
\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} = 3052.08 + 1525.72 = 4577.80 \, \text{cm}^3
\]

Answer for Problem 2:
\[
\boxed{4577.80}
\]

---

Problem 3:

The shape consists of a cube with a hemisphere on top.

#### Step 1: Volume of the Cube
The side length of the cube is 8 cm.

The formula for the volume of a cube is:
\[
V_{\text{cube}} = s^3
\]
\[
V_{\text{cube}} = 8^3 = 512 \, \text{cm}^3
\]

#### Step 2: Volume of the Hemisphere
The radius of the hemisphere is half the side length of the cube, which is 4 cm.

The formula for the volume of a hemisphere is:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\]
\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 4^3 = \frac{2}{3} \times 3.14 \times 64 = \frac{2}{3} \times 200.96 = 133.97 \, \text{cm}^3
\]

#### Step 3: Total Volume
The total volume is the sum of the volumes of the cube and the hemisphere:
\[
V_{\text{total}} = V_{\text{cube}} + V_{\text{hemisphere}} = 512 + 133.97 = 645.97 \, \text{cm}^3
\]

Answer for Problem 3:
\[
\boxed{645.97}
\]

---

Problem 4:

The shape consists of a rectangular prism with a triangular pyramid on top.

#### Step 1: Volume of the Rectangular Prism
The dimensions of the rectangular prism are:
- Length = 14 m
- Width = 9 m
- Height = 8 m

The formula for the volume of a rectangular prism is:
\[
V_{\text{prism}} = \text{Length} \times \text{Width} \times \text{Height}
\]
\[
V_{\text{prism}} = 14 \times 9 \times 8 = 1008 \, \text{m}^3
\]

#### Step 2: Volume of the Triangular Pyramid
The triangular pyramid has:
- Base (triangle) dimensions: Base = 14 m, Height = 9 m
- Height of the pyramid = 8 m

The area of the triangular base is:
\[
\text{Area of base} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 14 \times 9 = 63 \, \text{m}^2
\]

The formula for the volume of a pyramid is:
\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
\[
V_{\text{pyramid}} = \frac{1}{3} \times 63 \times 8 = \frac{1}{3} \times 504 = 168 \, \text{m}^3
\]

#### Step 3: Total Volume
The total volume is the sum of the volumes of the prism and the pyramid:
\[
V_{\text{total}} = V_{\text{prism}} + V_{\text{pyramid}} = 1008 + 168 = 1176 \, \text{m}^3
\]

Answer for Problem 4:
\[
\boxed{1176}
\]

---

Problem 5:

The shape consists of a rectangular prism with a smaller rectangular prism on top.

#### Step 1: Volume of the Larger Rectangular Prism
The dimensions of the larger rectangular prism are:
- Length = 4 mm
- Width = 3 mm
- Height = 2 mm

The formula for the volume of a rectangular prism is:
\[
V_{\text{larger prism}} = \text{Length} \times \text{Width} \times \text{Height}
\]
\[
V_{\text{larger prism}} = 4 \times 3 \times 2 = 24 \, \text{mm}^3
\]

#### Step 2: Volume of the Smaller Rectangular Prism
The dimensions of the smaller rectangular prism are:
- Length = 2 mm
- Width = 2 mm
- Height = 1 mm

The formula for the volume of a rectangular prism is:
\[
V_{\text{smaller prism}} = \text{Length} \times \text{Width} \times \text{Height}
\]
\[
V_{\text{smaller prism}} = 2 \times 2 \times 1 = 4 \, \text{mm}^3
\]

#### Step 3: Total Volume
The total volume is the sum of the volumes of the two prisms:
\[
V_{\text{total}} = V_{\text{larger prism}} + V_{\text{smaller prism}} = 24 + 4 = 28 \, \text{mm}^3
\]

Answer for Problem 5:
\[
\boxed{28}
\]

---

Problem 6:

The shape consists of a cone with a hemisphere on top.

#### Step 1: Volume of the Cone
The dimensions of the cone are:
- Radius = 12 cm
- Height = 13 cm

The formula for the volume of a cone is:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
\[
V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 12^2 \times 13 = \frac{1}{3} \times 3.14 \times 144 \times 13 = \frac{1}{3} \times 5873.28 = 1957.76 \, \text{cm}^3
\]

#### Step 2: Volume of the Hemisphere
The radius of the hemisphere is 12 cm.

The formula for the volume of a hemisphere is:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\]
\[
V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 12^3 = \frac{2}{3} \times 3.14 \times 1728 = \frac{2}{3} \times 5425.92 = 3617.28 \, \text{cm}^3
\]

#### Step 3: Total Volume
The total volume is the sum of the volumes of the cone and the hemisphere:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 1957.76 + 3617.28 = 5575.04 \, \text{cm}^3
\]

Answer for Problem 6:
\[
\boxed{5575.04}
\]

---

Final Answers:

\[
\boxed{1152, 4577.80, 645.97, 1176, 28, 5575.04}
\]