Have you ever honored someone with a gift? Take a look at this dilemma.

Mr. Samuels, the custodian, has been working overtime getting ready for the Olympics at Montgomery Middle School. To help the students, he has been creating long jump fields, bringing in tons of sand and offering to help out after school and on weekends.

“I think that we should do something nice for Mr. Samuels,” Crystal said at lunch.

“I agree. But what?” Kenneth responded.

“How about giving him one of our prism awards? I could wrap it up and we could present it to him at the Olympics,” Crystal suggested.

Kenneth, Marcy and Dylan all agreed. So that week, Crystal found a box for the prism award and began to wrap it. Because of the size of the award, she needed a good sized box.

How much wrapping paper will she need?

**
Surface area is the topic for this Concept, and Crystal’s wrapping paper dilemma is all about surface area. By the end of this Concept, you will know how much wrapping paper she will need to cover this box.
**

### Guidance

**
A three–dimensional or solid figure has length, width and depth.
**

This Concept focuses on prisms and surface area.

**
What is a prism?
**

A prism is a three–dimensional figure with two parallel congruent polygons as bases. The side faces of a prism are rectangular in shape.

**
One of the things that we can measure when working with three–dimensional figures is called
**
**
surface area
**

**. Surface area is the total of the areas of each face of a solid figure. Imagine you could wrap one of the figures above in wrapping paper, like a present.**

The amount of wrapping paper needed to cover the figure represents its surface area. To find the surface area, we must be able to calculate the area of each face and then add these areas together.

There are several different ways to calculate surface area.

One way is to use a
**
net
**
.

**
A net is a two-dimensional diagram of a three-dimensional figure.
**
Imagine you could unfold a box so that it is completely flat. You would have something that looks like this.

If we folded this up, you could see that it would form a cube. A cube is made up of faces that are squares. If we wanted to figure out the surface area or measurement of the outer covering of this cube, then we could find the area of each surface of the cube and then add the products together.

We could also look at a net of a rectangular prism.

A rectangular prism is made up of rectangles. To find the surface area of a prism, we would need to calculate the area of each of the faces and then add them together.

**
Let’s begin by calculating the surface area of a rectangular prism.
**

**
Now that we have all the information we need, we can calculate the area of each face and then add their areas together.
**

\begin{align*}& \mathbf{bottom \ face} && \mathbf{top \ face} && \mathbf{long \ side} && \mathbf{long \ side} && \mathbf{short \ side} && \mathbf{short \ side}\\ & A=lw && A=lw && A=lh && A=lh && A=wh && A=wh\\ & 12 \times 7 \quad + && 12 \times 7 \quad + && 12 \times 3 \quad + && 12 \times 3 \quad + && 7 \times 3 \quad + && 7 \times 3\\ & 84 \qquad \ \ + && 84 \qquad \ \ + && 36 \qquad \ \ + && 36 \qquad \ \ + && 21 \qquad \ + && 21 \qquad \ \ = \ 282 \ in.^2\end{align*}

**
We found the area of each rectangular face and then added all of these areas together. The total surface area of the rectangular prism is 282 square inches. Using a net helped us to locate all of the faces and find the measurements of each side.
**

Nets let us see each face so that we can calculate their area and then add them together. However, we can also use a formula to represent the faces as we find their area. The formula gives us a nice short cut that we can use for any kind of prism, no matter what shape its base is. Take a look at the formula below.

@$\begin{align*}SA=Ph+2B\end{align*}@$

**
Let’s look at the first part of the formula.
@$\begin{align*}P\end{align*}@$
represents the perimeter of the base, and
@$\begin{align*}h\end{align*}@$
represents the height of the prism. By multiplying the perimeter and height, we are finding the area of all of the side faces at once. This will be very useful if the prism that we are working with isn’t just a cube or a rectangular prism.
**

**
The second part of the formula represents the area of the top and bottom faces.
@$\begin{align*}B\end{align*}@$
represents the area of one base, which we find using whichever area formula is appropriate for the shape of the base. Then we multiply it by 2 to show the area of the top and bottom faces at once. Let’s give it a try to see how this works.
**

**
Find the surface area of this figure using a formula.
**

**
We have all the measurements we need. Let’s find the perimeter of the base first. It is a rectangle, so we add the lengths and widths:
@$\begin{align*}21 + 21 + 14 + 14 = 70\end{align*}@$
. We can put this number in for
@$\begin{align*}P\end{align*}@$
in the formula. The height, we can see, is 5 centimeters.
**

**
Now let’s solve for
@$\begin{align*}B\end{align*}@$
, the area of the base. The base of this prism is a rectangle, so we use the formula
@$\begin{align*}A = lw\end{align*}@$
to find its area.
**

@$$\begin{align*}B &= lw\\ B &=21 \times 14\\ B &=294 \ cm^2\end{align*}@$$

**
Now we have all of the information we need to fill in the formula. Let’s put it in and solve for
@$\begin{align*}SA\end{align*}@$
, surface area.
**

@$$\begin{align*}SA &=Ph + 2B\\ SA &= 70 (5) + 2 (294)\\ SA &= 350 + 588\\ SA &= 938 \ cm^2\end{align*}@$$

**
This rectangular prism has a surface area of 938 square centimeters.
**

**
Write this formula for finding the surface area of a prism down in your notebook.
**

Answer each question.

#### Example A

True or false. The surface area includes the inside of a prism.

**
Solution: False. The surface area is the measurement of the outer covering of a prism.
**

#### Example B

True or false. A net shows all three dimensions of a prism.

**
Solution: True.
**

#### Example C

True or false. You know a figure is a prism because the faces are rectangles.

**
Solution: True
**

Now let's go back to the dilemma from the beginning of the Concept.

**
First of all, what kind of solid figure is this? All of the faces are rectangles, including the base, so it is a rectangular prism. The picture clearly shows us what its length, width, and height are, so let’s use the formula for finding the surface area of prisms.
**

**
What is the perimeter of the base?
**

@$\begin{align*}12 + 12 + 9 + 9 = 42 \ inches\end{align*}@$ .

**
We’ll put this in for
@$\begin{align*}P\end{align*}@$
.
**

**
We also need to find the area of the base,
@$\begin{align*}B\end{align*}@$
. This base is a rectangle, so we use the formula
@$\begin{align*}B = lw\end{align*}@$
.
**

@$$\begin{align*}B &=lw \\ B &=12 (9)\\ B &=108 \ in.^2\end{align*}@$$

**
Now we have all of the measurements to put in for the appropriate variables in the formula.
**

@$$\begin{align*}SA &=Ph + 2B\\ SA &=42(6)+2(108)\\ SA &=252 + 216\\ SA &=468 \ in.^2\end{align*}@$$

**
Crystal will need 468 square inches of wrapping paper in order to cover the present.
**

### Guided Practice

Here is one for you to try on your own.

**
What is the surface area of the figure below?
**

**
Solution
**

Let’s look at the base first to find its perimeter. The triangle has two sides of 5 inches and one that is 8 inches: @$\begin{align*}5 + 5 + 8 = 18 \ inches\end{align*}@$ . This will be @$\begin{align*}P\end{align*}@$ in the formula. The height of the prism is 15 inches. Be careful not to confuse the height of the prism with the height of the triangular base!

To find @$\begin{align*}B\end{align*}@$ , we need to use the area formula for triangles: @$\begin{align*}A =\frac{1}{2} bh\end{align*}@$ . The base of the triangle is 8 inches, and the height is 3 inches.

@$$\begin{align*}A &=\frac{1}{2} bh\\ A &= \frac{1}{2} (8) (3)\\ A &= 4 (3)\\ A &= 12 \ in.^2\end{align*}@$$

The area of the triangular base is 12 square inches, so we put this in for @$\begin{align*}B\end{align*}@$ in the formula. Let’s put all of the values in and solve.

@$$\begin{align*}SA &= Ph + 2B\\ SA &= 18 (15) + 2 (12)\\ SA &= 270 + 24\\ SA &= 294 \ in.^2\end{align*}@$$

**
The surface area of this triangular prism is 294 square inches.
**

### Video Review

### Explore More

Directions: Look at each figure and then answer the following questions about each.

- What is the name of the figure pictured above?
- What is the surface area of this figure?
- What is the shape of it's base?
- What is the height of this figure?
- What is the area of this figure's base?

- What is the name of this figure?
- What is the surface area of this figure?
- What is the shape of it's base?
- What is the height of this figure?
- What is the area of this figure's base?

- What is the name of this figure?
- What is the shape of the base
- How many bases does this figure have?
- How many side faces are there?
- What is the surface area of this figure?