Volume of Prisms and Cylinders worksheet with nine geometric figures and space to calculate volume.

To solve the problem of finding the volumes of the given prisms and cylinders, we will use the appropriate volume formulas for each shape. Let's go through each figure step by step.

1) Rectangular Prism

The formula for the volume of a rectangular prism is:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Length = 5 ft
- Width = 5 ft
- Height = 10 ft

\[
V = 5 \times 5 \times 10 = 250 \, \text{ft}^3
\]

2) Cylinder

The formula for the volume of a cylinder is:
\[
V = \pi r^2 h
\]
Given:
- Radius \( r = 8 \, \text{ft} \)
- Height \( h = 12 \, \text{ft} \)

\[
V = \pi (8)^2 (12) = \pi \times 64 \times 12 = 768\pi \approx 2412.74 \, \text{ft}^3
\]

3) Cube

The formula for the volume of a cube is:
\[
V = s^3
\]
Given:
- Side length \( s = 6 \, \text{in} \)

\[
V = 6^3 = 216 \, \text{in}^3
\]

4) Rectangular Prism

The formula for the volume of a rectangular prism is:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Length = 10 cm
- Width = 6 cm
- Height = 3 cm

\[
V = 10 \times 6 \times 3 = 180 \, \text{cm}^3
\]

5) Triangular Prism

The formula for the volume of a triangular prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
The base is a triangle with:
- Base = 6 yd
- Height = 5 yd

First, find the area of the triangular base:
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 5 = 15 \, \text{yd}^2
\]

Now, multiply by the height of the prism:
\[
V = 15 \times 13 = 195 \, \text{yd}^3
\]

6) Pentagonal Prism

The formula for the volume of a pentagonal prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
The base is a regular pentagon with side length 3 in and apothem 2 in.

First, find the area of the pentagonal base:
\[
\text{Base Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
The perimeter of the pentagon is:
\[
\text{Perimeter} = 5 \times 3 = 15 \, \text{in}
\]

\[
\text{Base Area} = \frac{1}{2} \times 15 \times 2 = 15 \, \text{in}^2
\]

Now, multiply by the height of the prism:
\[
V = 15 \times 10 = 150 \, \text{in}^3
\]

7) Hexagonal Prism

The formula for the volume of a hexagonal prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
The base is a regular hexagon with side length 7 mm and height 13 mm.

First, find the area of the hexagonal base:
\[
\text{Base Area} = \frac{3\sqrt{3}}{2} s^2
\]
where \( s = 7 \, \text{mm} \).

\[
\text{Base Area} = \frac{3\sqrt{3}}{2} \times 7^2 = \frac{3\sqrt{3}}{2} \times 49 = \frac{147\sqrt{3}}{2} \approx 127.31 \, \text{mm}^2
\]

Now, multiply by the height of the prism:
\[
V = 127.31 \times 13 \approx 1655.03 \, \text{mm}^3
\]

8) Cylinder

The formula for the volume of a cylinder is:
\[
V = \pi r^2 h
\]
Given:
- Radius \( r = 6 \, \text{cm} \)
- Height \( h = 8 \, \text{cm} \)

\[
V = \pi (6)^2 (8) = \pi \times 36 \times 8 = 288\pi \approx 904.78 \, \text{cm}^3
\]

9) Triangular Prism

The formula for the volume of a triangular prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
The base is a right triangle with:
- Base = 3 yd
- Height = 2 yd

First, find the area of the triangular base:
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 2 = 3 \, \text{yd}^2
\]

Now, multiply by the height of the prism:
\[
V = 3 \times 10 = 30 \, \text{yd}^3
\]

Final Answers

\[
\boxed{
\begin{array}{ll}
1) & 250 \, \text{ft}^3 \\
2) & 2412.74 \, \text{ft}^3 \\
3) & 216 \, \text{in}^3 \\
4) & 180 \, \text{cm}^3 \\
5) & 195 \, \text{yd}^3 \\
6) & 150 \, \text{in}^3 \\
7) & 1655.03 \, \text{mm}^3 \\
8) & 904.78 \, \text{cm}^3 \\
9) & 30 \, \text{yd}^3 \\
\end{array}
}
\]