### Let’s Think About It

Crystal has a fragile gift that she is packing in a box before wrapping it. To figure out how much wrapping paper she will need she has drawn the box with all its dimensions. How can Crystal calculate how much gift-wrap she will need using the above dimensions?

In this concept, you will learn to find the surface area of prisms.

### Guidance

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

You 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, you 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.

First, let’s break apart the shape so you can organize the six area calculations.

Next, calculate the area of each face.

The answer is 282.

The surface area of the prism is

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

Take a look at the formula below.

Let’s look at the first part of the formula.

represents the perimeter of the base, and represents the height of the prism. By multiplying the perimeter and height, you are finding the area of all of the side faces at once. This will be very useful if the prism that you 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.

represents the area of one base, which you find using whichever area formula is appropriate for the shape of the base. Then you multiply it by 2 to show the area of the top and bottom faces at once.Let’s look at an example.

Find the surface area of this figure using a formula.

First, find the perimeter of the base.

Next, calculate the area of the base.

Then, knowing the height is 5 cm, determine the surface area.

The answer is 938.

The surface area of the prism is

.### Guided Practice

What is the surface area of the figure below?

First, find the perimeter of the base.

Next, calculate the area of the base.

Then, knowing the height is 15 in, determine the surface area.

The answer is 294.

The surface area of the prism is

.### Examples

#### Example 1

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

The surface area is the measurement of the outer covering of a prism.

The answer is false.

#### Example 2

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

A net does show all dimensions of a prism.

The answer is true.

#### Example 3

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

A figure with all rectangular faces is a prism.

The answer is true.

### Follow Up

Remember Crystal and her gift box?

The box to be wrapped has a length of

, a wide of and a height of .First, find the perimeter of the base.

Next, calculate the area of the base.

Then, knowing the height is 6 in., determine the surface area.

The answer is 468.

Crystal needs

of wrapping paper.### Video Review

https://www.youtube.com/watch?v=WJ7rYIAdAbA

### Explore More

Look at each figure and then answer the following questions.

1. What is the name of the figure pictured above?

2. What is the surface area of this figure?

3. What is the shape of it’s base?

4. What is the height of this figure?

5. What is the area of this figure’s base?

6. What is the name of this figure?

7. What is the surface area of this figure?

8. What is the shape of it’s base?

9. What is the height of this figure?

10. What is the area of this figure’s base?

11. What is the name of this figure?

12. What is the shape of the base?

13. How many bases does this figure have?

14. How many side faces are there?

15. What is the surface area of this figure?