# 10.16: Volume of Prisms

**At Grade**Created by: CK-12

**Practice**Volume of Prisms

Let's take another look at Jillian's box.

In the last Concept, you learned how to count unit cubes to figure out the volume of different prisms. Well, there is an easier way. We can use a formula to calculate the volume of a prism. Here are the dimensions of Jillian's box once again.

Jillian's box is a rectangular prism and has the following dimensions: 7" x 6" x 4".

How can we use a formula to calculate the volume of this prism?

**Pay attention and this Concept will teach you all that you need to know.**

### Guidance

Looking at all of those cubes is a simple, easy way to understand volume. If you can count the cubes, you can figure out the volume. However, not all of the prisms that you will work with will have the cubes drawn in. In this Concept, you will learn how to figure out the volume of a prism when there aren’t any cubes drawn inside it.

**How can we figure out the volume of a prism without counting cubes?**

**Here we have the dimensions written on a rectangular prism. This prism has a height of 5 inches, a width of three inches and a length of four inches.**

You can see that a few cubes have been drawn in to show you that if we continued filling the cubes that they would be four cubes across by three cubes wide, and we would build them five cubes high.

**That’s right! Here is how it works.**

\begin{align*}V = Bh\end{align*}

\begin{align*}B\end{align*} means the area of the base and \begin{align*}h\end{align*} means the height.

The area of the base is length times width.

\begin{align*}A & = 3 \times 4 = 12\\ h & = 5\\ V & = 12 \times 5 = 60\end{align*}

**The volume is 60 cubic inches or \begin{align*}in^3\end{align*}.**

\begin{align*}V = Bh\end{align*}

The area of the base is 2 \begin{align*}\times\end{align*} 8 = 16

The height is 3 inches.

\begin{align*}V & = 16 \times 3\\ V & = 48 \ in^3\end{align*}

**The volume of this rectangular prism is \begin{align*}48 \ in^3\end{align*}.**

**How can we find the volume of a triangular prism?**

We can use the same formula for finding the volume of the triangular prism. Except this time, the area of the base is a triangle and not a rectangle.

\begin{align*}V = Bh\end{align*}

To find the volume of a triangular prism, we multiply the area of the base \begin{align*}(B)\end{align*} with the height of the prism.

To find the area of a triangular base we use the formula for area of a triangle.

\begin{align*}A & = \frac{1}{2}bh\\ A & = \frac{1}{2}(15 \times 6)\\ A & = \frac{1}{2}(90)\\ A & = 45 \ sq. \ units\\ V & = Bh\\ V & = (45)h\\ V & = 45(2)\\ V & = 90 \ cubic \ centimeters \ or \ cm^3\end{align*}

**The volume of the prism is \begin{align*}90 \ cm^3\end{align*}.**

Now that you know how to find the volume of prisms using a formula, it is time to practice.

#### Example A

**Solution: \begin{align*}125 in^3\end{align*}**

#### Example B

**Solution: \begin{align*}450 in^3\end{align*}**

#### Example C

**Solution: \begin{align*} 17.5 cm^3\end{align*}**

Do you know how to use the formula for finding the volume of a prism? Here is the original problem once again.

Let's take another look at Jillian's box.

In the last Concept, you learned how to count unit cubes to figure out the volume of different prisms. Well, there is an easier way. We can use a formula to calculate the volume of a prism. Here are the dimensions of Jillian's box once again.

Jillian's box is a rectangular prism and has the following dimensions: 7" x 6" x 4".

How can we use a formula to calculate the volume of this prism?

\begin{align*}V = Bh\end{align*}

Now we can substitute in the given values for length, width and height.

\begin{align*}V = (7 \times 6)(4)\end{align*}

**The volume of Jillian's box is \begin{align*} 168 in^3\end{align*}.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Surface area
- the outer covering of a solid figure-calculated by adding up the sum of the areas of all of the faces and bases of a prism.

- Net
- diagram that shows a “flattened” version of a solid. Each face and base is shown with all of its dimensions in a net. A net can also serve as a pattern to build a three-dimensional solid.

- Triangular Prism
- a solid which has two congruent parallel triangular bases and faces that are rectangles.

- Rectangular Prism
- a solid which has rectangles for bases and faces.

- Volume
- the amount of space inside a solid figure

### Guided Practice

Here is one for you to try on your own.

**Answer**

To find the volume of a prism, we use the following formula.

\begin{align*}V = Bh\end{align*}

Now we substitute in the given values.

\begin{align*}V = (16 \times 9)(4)\end{align*}

\begin{align*}V = 576 cm^3\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

Khan Academy: Solid Geometry Volume

### Practice

Directions: Find the volume of each rectangular prism. Remember to label your answer in cubic units.

1. Length = 5 in, width = 3 in, height = 4 in

2. Length = 7 m, width = 6 m, height = 5 m

3. Length = 8 cm, width = 4 cm, height = 9 cm

4. Length = 8 cm, width = 4 cm, height = 12 cm

5. Length = 10 ft, width = 5 ft, height = 6 ft

6. Length = 9 m, width = 8 m, height = 11 m

7. Length = 5.5 in, width = 3 in, height = 5 in

8. Length = 6.6 cm, width = 5 cm, height = 7 cm

9. Length = 7 ft, width = 4 ft, height = 6 ft

10. Length = 15 m, width = 8 m, height = 10 m

Directions: Find the volume of each triangular prism. Remember that \begin{align*}h\end{align*} means the height of the triangular base and \begin{align*}H\end{align*} means the height of the whole prism.

11. \begin{align*}b = 6 \ in, \ h = 4 \ in, \ H = 5 \ in\end{align*}

12. \begin{align*}b = 7 \ in, \ h = 5 \ in, \ H = 9 \ in\end{align*}

13. \begin{align*}b = 10 \ m, \ h = 8 \ m, \ H = 9 \ m\end{align*}

14. \begin{align*}b = 12 \ m, \ h = 10 \ m, \ H = 13 \ m\end{align*}

15. \begin{align*}b = 8 \ cm, \ h = 6 \ cm, \ H = 9 \ cm\end{align*}

Directions: Answer true or false for each of the following questions.

16. Volume is the amount of space that a figure can hold inside it.

17. The volume of a rectangular prism is always greater than the volume of a cube.

18. The volume of a triangular prism is less than a rectangular prism with the same size base.

19. A painter would need to know the surface area of a house to do his/her job correctly.

20. If Marcus is covering his book with a book cover, Marcus is covering the surface area of the book.

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.Rectangular Prism

A rectangular prism is a prism made up of two rectangular bases and four rectangular faces.Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.Triangular Prism

A triangular prism is a prism made up of two triangular bases and three rectangular faces.### Image Attributions

Here you'll learn to find volumes of rectangular and triangular prisms using formulas.

## Concept Nodes:

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.Rectangular Prism

A rectangular prism is a prism made up of two rectangular bases and four rectangular faces.Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.Triangular Prism

A triangular prism is a prism made up of two triangular bases and three rectangular faces.