# 10.6: Surface Area and Volume of Prisms

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

## Introduction

*The Sewing Box*

Now that Jillian knows that her sewing box is a prism, she needs to figure out how much material she will need for the entire box.

Jillian needs to figure out the surface area of the box, so she will be able to figure out how much material she will need. Jillian can’t remember how to figure out the surface area, but she knows that she will need a sketch. Here is her drawing.

She knows that the measurements are 7” long \begin{align*}\times\end{align*}

**This is where you come in. In this lesson, you will learn about finding the surface area and volume of different prisms. Pay close attention and by the end of the lesson you will know how to help Jillian with her dilemma.**

*What You Will Learn*

By the end of this lesson, you will be able to demonstrate the following skills.

- Identify surface area of prisms as the sum of the areas of faces using nets.
- Find surface area of rectangular and triangular prisms using formulas.
- Identify volumes of prisms as the sum of volumes of layers of unit cubes.
- Find volumes of rectangular and triangular prisms using formulas.

*Teaching Time*

I. **Identify Surface Area of Prisms as the Sum of the Areas of Faces Using Nets**

When you learned about plane figures such as rectangles and squares, you learned how to calculate the area of the figure.

Example

This rectangle has a length of 8 inches and a width of 3 inches. You can find the area of a rectangle by multiplying the length times the width.

8 \begin{align*}\times\end{align*}

**The area of the rectangle is 24 sq. in or \begin{align*}in^2\end{align*} in2.**

**What is surface area?**

** Surface area** is the total area of each of the faces of a three-dimensional object.

Let’s look at a cube and see how this works.

**Let’s say that we wanted to find the surface area of the cube.**

**What would that be exactly?**

**The surface area of the cube would be the total area of all of the blue surfaces.**

Finding the surface area of a figure is very useful when painting or covering a three-dimensional solid. You have to know the total area of the whole solid to know how much paint or cloth or covering you are going to need.

**How can we calculate the surface area of a figure?**

**We figure out the surface area by calculating the area of each of the faces of the solid and then add up all of the areas for the total surface area.**

**That is a good question! If you look at the cube we just looked at, it is hard to see all of the sides.**

**However, we can use a** *net***to see all of the sides of a three-dimensional solid.**

**A net is a drawing that shows a “flattened out” picture of the solid. With a net we can see each part of the solid. If you were to make a net out of paper and fold it up, you would be able to create a solid figure.**

Here is a net of a cube.

**You learned in the last lesson that a cube has six faces. Well, you can see here that this net also has six faces. If you were to fold this figure using the line segments you would see that it would create a cube.**

**How do we use a net to calculate surface area?**

**To calculate surface area, we find the area of each of the six faces of the cube and then add up all of the areas.**

*It is a bit simpler with a cube because each square side is the same size. It is more challenging to work with a rectangular prism. However, we will try an example with a cube first to get the idea.*

Example

The length of one of the sides of the square face is 3 inches. We can use the formula for finding the area of a square to find the area of one square.

\begin{align*}A & = s^2\\
A & = 3^2\\
A & = 9 \ sq. \ in.\end{align*}

**This is the area for one square face. There are six square faces. If we take this area and multiply it by six or add the area six times, we will have the surface area of the cube.**

\begin{align*}9(6) & = 54 \ sq. \ in.\\
9 + 9 + 9 + 9 + 9 + 9 & = 54 \ in^2\end{align*}

**The surface area of the cube is \begin{align*}54 \ in^2\end{align*} 54 in2.**

**Now let’s look at an example of a rectangular prism net.**

Let’s say that this box has a length of 6”, a width of 4” and a height of 2”.

**We need to find the area of each rectangle.**

**There are two long faces.**

**There are two short faces.**

**There is one bottom, and one top that matches it.**

**First, let's find the area of the bottom. It has a length of 6” and a width of 4”. Since the shape of the bottom is a rectangle, we can use the formula for finding the area of a rectangle.**

\begin{align*}A & = lw\\
A & = (6)(4)\\
A & = 24 \ sq. \ in.\end{align*}

Since the top and bottom match, we can multiply this area by two: 24 x 2 = 48 square inches.

**Next, we find the area of the two long faces. Each long face is a rectangle in shape. The length is 6” and the width is 2”.**

\begin{align*}A & = lw\\
A & = (6)(2)\\
A & = 12 \ sq. \ in.\end{align*}

**Since there are two long faces to the prism, we can take this measure and multiply it by two.**

\begin{align*}A = 24 \ sq. \ in\end{align*}

**Next, we find the area of the two shorter faces. Each face is small rectangle. The length is 4” and the width is 2”.**

\begin{align*}A & = lw\\
A & = (4)(2)\\
A & = 8 \ sq. \ in.\end{align*}

**Since there are two short faces to the prism, we can take this measure and multiply it by two.**

\begin{align*}A & = 2(8)\\
A & = 16 \ sq. \ in\end{align*}

**To find the surface area of the entire prism, we add up the areas of all of the faces.**

\begin{align*}SA = 16 + 24 + 48 = 88 \ sq. \ inches\end{align*}

**This is our answer.**

**What about a triangular prism?**

A ** triangular prism** is a prism with two parallel congruent bases that are triangles. The faces of the prism are rectangles, but the bases are triangles. Here is a picture of a triangular prism.

Here is what the net of a triangular prism would look like.

We need to figure out the area of the bottom, right side, left side and two bases which are triangles.

**The bottom is a rectangle. It has a length of 7 cm and a width of 3 cm.**

\begin{align*}A & = lw\\
A & = 7(3)\\
A & = 21 \ sq. \ cm.\end{align*}

**The left side is a rectangle. It has a length of 7 cm and a width of 4 cm.**

\begin{align*}A & = lw\\
A & = 7(4)\\
A & = 28 \ sq. \ cm.\end{align*}

**The right side is a rectangle. It has a length of 7 cm and a width of 5 cm.**

\begin{align*}A & = lw\\
A & = 7(5)\\
A & = 35 \ sq. \ cm.\end{align*}

**The bases are two triangles. They have a base of 3 cm and a height of 4 cm.**

\begin{align*}A & = \frac{1}{2}bh\\
A & = \frac{1}{2}(3)(4)\\
A & = \frac{1}{2}(12)\\
A & = 6 \ sq. \ cm\end{align*}

**There are two triangles, so we can multiply this base by two.**

\begin{align*}A & = 2(6)\\
A & = 12 \ sq. \ cm.\end{align*}

**Now we add up all of the areas.**

\begin{align*}SA & = 12 + 35 + 28 + 21\\
SA & = 96 \ sq. \ cm.\end{align*}

**Now practice finding the surface area of a prism on your own. Figure out the surface area of each prism.**

*Take a few minutes to check your work with a partner.*

II. **Find the Surface Area of Rectangular and Triangular Prisms Using Formulas**

In the last section we figured out the surface area of rectangular and triangular prisms using nets. We can also use formulas to figure out surface area. In fact, often times, you won’t have a net to work with. You can always draw one, but if you know which formula to use, you can figure out the surface area of the prism using a formula.

**How can we figure out the surface area of a rectangular prism without a net?**

We can figure out the surface area of a rectangular prism by using a formula. Let’s look at a diagram and then a formula to find the surface area of the rectangular prism.

To find the surface area of this rectangular prism, we have to figure out the sum of all of the areas. Here is a formula that we can use to do this.

\begin{align*}SA = 2(lw + lh + wh)\end{align*}

We can substitute the given values into the formula. The length of the prism is 9 inches, the width is 3 inches and the height is 5 inches.

\begin{align*}SA & = 2(9(3) + 9(5)+ (3)5)\\
SA & = 2(27 + 45 + 15)\\
SA & = 2(87)\\
SA & = 174 \ sq. \ in.\end{align*}

We can do this same work with a triangular prism. Let’s look at a diagram and a formula to find the surface area of a triangular prism.

\begin{align*}SA & = Area \ of \ three \ rectangles + Area \ of \ two \ triangles\\ SA & = 2(8 + 9 + 7) + 2\left (\frac{1}{2}(8)7\right )\\ SA & = 2(24) + 2(28)\\ SA & = 48 + 56\\ SA & = 104 \ sq. \ in.\end{align*}

**Now it’s time to practice. Find the surface area of each prism using a formula.**

*Take a few minutes to check your work with a partner.*

III. **Identify Volumes of Prisms as the Sum of Volumes of Layers of Unit Cubes**

In this section, we will look at the ** volume** of prisms.

**Volume is the amount of space inside a solid figure**. In this section, we will look at the volume of prisms.

**Let’s start with an example.**

**These cubes make up a rectangular prism. The cubes represent the volume of the prism.**

This prism is five cubes by two cubes by one cube. In other words, it is five cubes long, by two cubes high by one cube wide.

We can multiply each of these values together to get the volume of the rectangular prism.

5 \begin{align*}\times\end{align*} 2 \begin{align*}\times\end{align*} 1 = 10 cubic units

If we count the cubes, we get the same result.

**The volume of the rectangular prism is 10 cubic units or units\begin{align*}^3\end{align*}.**

**Identify the volume of each prism.**

*Check your answers. Do you notice anything about the solid in question1 and the solid in question 3?*

IV. **Find Volumes of Rectangular and Triangular Prisms Using Formulas**

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 section, 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?**

To understand how this works, let’s look at an example.

**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*}.**

**Let’s look at another example without the cubes drawn in.**

Example

\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. Let’s look at an example.

Example

\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.**

*Did you end up with a decimal answer for question 3? Did you label your work in cubic units? Check your answers with a friend before moving on.*

## Real Life Example Completed

*The Sewing Box*

**Here is the original problem once again. You have learned all about volume and surface area and should be ready to help Jillian. Reread the problem and underline any important information.**

Now that Jillian knows that her sewing box is a prism, she needs to figure out how much material she will need for the entire box.

Jillian needs to figure out the surface area of the box. If she can figure out the surface area, she will be able to figure out how much material she will need. Jillian can’t remember how to figure out the surface area, but she knows that she will need a sketch. Here is her drawing.

She knows that the measurements are 7” long \begin{align*}\times\end{align*} 6” wide by 4” high. Jillian isn’t sure which measurements go where.

**Jillian wants to label the exact dimensions of the box. To do this, she has to think of it as a put together box. If she does this, she will notice that the length is the long part of the box, the width is the next longest piece and the height is the tiny piece of the sides. When this is put together, Jillian will see all of the parts of the box quite clearly.**

**To figure out the surface area of the box so that Jillian will know how much material to buy, Jillian can use the formula for finding the surface area of a rectangular prism.**

\begin{align*}SA & = 2(lw + lh + wh)\\ SA & = 2(7(6) + 7(4) + (4)6)\\ SA & = 2(42 + 28 + 24)\\ SA & = 188 \ square \ inches\end{align*}

**Jillian will need to purchase 188 square inches of material.**

Since this material is in square inches, Jillian will purchase a square that has an area of at least 188 sq. inches. To figure out how big a square to purchase, Jillian works to think of a number that multiplied by itself is equal to at least 188 square inches. She starts with 12.

\begin{align*}12” \times 12” & = 144 \ sq. \ in && Nope-too \ small\\ 13” \times 13” & = 169 \ sq. \ in && Nope-still \ too \ small\\ 14” \times 14” & = 196 \ sq. \ in\end{align*}

*Perfect! Jillian will have plenty of material and there will be some left over too just in case of a mistake!!*

## Vocabulary

Here are the vocabulary words that can be found in this lesson.

- 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

## Technology Integration

- http://www.mathplayground.com/mv_surface_area_prisms.html – This is a Brightstorm video on the surface area of prisms.

- http://www.mathplayground.com/mv_volume_prisms.html – This is a Brightstorm video on finding the volume of a prism.

## Time to Practice

Directions: Find the surface area and volume of each prism.

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Directions: Identify each type of prism.

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

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

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

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

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

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

20. The amount of water in a pool is the volume of the pool.