Volume of Prisms and Cylinders Worksheet | PDF | Volume | Space - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1.) Rectangular Prism

Find Area of Base and Volume.

* Step 1: Identify the base. The base is the rectangle at the bottom with length $14.5 \text{ m}$ and width $3 \text{ m}$.
* Step 2: Calculate Area of Base.
$$\text{Area} = \text{length} \times \text{width}$$
$$\text{Area} = 14.5 \times 3 = 43.5 \text{ m}^2$$
* Step 3: Calculate Volume.
$$\text{Volume} = \text{Area of Base} \times \text{height}$$
The height is $5 \text{ m}$.
$$\text{Volume} = 43.5 \times 5 = 217.5 \text{ m}^3$$

2.) Triangular Prism

Find Area of Base and Volume.

* Step 1: Identify the base. The base is the triangle on the side. It has a base of $8 \text{ m}$ and a height of $6 \text{ m}$.
* Step 2: Calculate Area of Base.
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
$$\text{Area} = 0.5 \times 8 \times 6 = 24 \text{ m}^2$$
* Step 3: Calculate Volume.
$$\text{Volume} = \text{Area of Base} \times \text{length of prism}$$
The length connecting the triangles is $12 \text{ m}$. (Note: The $5.5 \text{ m}$ and $8 \text{ m}$ sides are not needed for volume).
$$\text{Volume} = 24 \times 12 = 288 \text{ m}^3$$

3.) Cylinder

Find Area of Base and Volume.

* Step 1: Identify dimensions. The diameter is $12 \text{ ft}$, so the radius ($r$) is half of that: $6 \text{ ft}$. The height ($h$) is $9 \text{ ft}$.
* Step 2: Calculate Area of Base.
$$\text{Area} = \pi \times r^2$$
$$\text{Area} = \pi \times 6^2 = 36\pi \approx 113.10 \text{ ft}^2$$
* Step 3: Calculate Volume.
$$\text{Volume} = \text{Area of Base} \times \text{height}$$
$$\text{Volume} = 36\pi \times 9 = 324\pi \approx 1,017.88 \text{ ft}^3$$

4.) Rectangular Prism

Find Area of Base and Volume.

* Step 1: Identify the base. The base is the rectangle at the bottom with dimensions $5 \text{ in}$ and $3 \text{ in}$.
* Step 2: Calculate Area of Base.
$$\text{Area} = 5 \times 3 = 15 \text{ in}^2$$
* Step 3: Calculate Volume.
$$\text{Volume} = \text{Area of Base} \times \text{height}$$
The height is $18 \text{ in}$.
$$\text{Volume} = 15 \times 18 = 270 \text{ in}^3$$

5.) Find Missing Dimension

Given Volume = $135 \text{ ft}^3$. Find $x$.

* Step 1: Set up the volume formula.
$$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$
$$135 = 15 \times 1.5 \times x$$
* Step 2: Multiply the known numbers.
$$15 \times 1.5 = 22.5$$
So, $135 = 22.5 \times x$
* Step 3: Solve for $x$.
$$x = \frac{135}{22.5}$$
$$x = 6 \text{ ft}$$

6.) Find Missing Dimension

Given Volume = $4,069.44 \text{ ft}^3$. Find radius $x$.

* Step 1: Set up the volume formula for a cylinder.
$$\text{Volume} = \pi \times r^2 \times h$$
$$4,069.44 = \pi \times x^2 \times 16$$
* Step 2: Divide by $\pi$ and the height to isolate $x^2$.
First, divide by $16$:
$$\frac{4,069.44}{16} = 254.34$$
Now, divide by $\pi$ (using $3.14159...$):
$$x^2 = \frac{254.34}{\pi} \approx 81$$
* Step 3: Take the square root.
$$x = \sqrt{81}$$
$$x = 9 \text{ ft}$$

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Final Answer:

1.)
Area of Base: 43.5 m²
Volume: 217.5 m³

2.)
Area of Base: 24 m²
Volume: 288 m³

3.)
Area of Base: 113.10 ft² (or $36\pi$)
Volume: 1,017.88 ft³ (or $324\pi$)

4.)
Area of Base: 15 in²
Volume: 270 in³

5.)
Missing dimension ($x$): 6 ft

6.)
Missing dimension ($x$): 9 ft