Leola owns an upholstery shop that specializes in refurbishing old furniture. A customer brought in an old wooden chest and wants Leola to cover the entire chest in a new vinyl covering. Leola needs to find the total surface area of the chest so she can order the vinyl covering. The chest is a rectangular prism with a net pictured below.

How can Leola figure out the total surface area of the chest?

In this concept, you will learn to identify the surface area of prisms by using nets and formulas.

### Finding Surface Area of Prisms

You can compute the area of a two-dimensional flat plane, like the rectangle below, by multiplying the length times the width.

This rectangle has a length of 8 inches and a width of 3 inches.

\begin{align*}8 \ in. \times 3 \ in. = 24 \ sq. in\end{align*}

The area of the rectangle is

or .When it comes to solid figures, you do not just calculate the area, you have to calculate the total **surface area**, which 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.

The surface area of this cube would be the total area of all of its surfaces. Finding the surface area of a figure is very useful when painting or covering a three-dimensional solid.

To figure out the surface area, calculate the area of each of the faces of the solid and then add all of the areas for the total surface area. Sometimes, it is hard to see all the sides of a solid object, so a **net** is used to see all the sides of a three-dimensional solid.

A **net** is a drawing that shows a “flattened out” picture of the solid. With a net, you 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.

A cube has six faces – two bases and four sides. Well, you can see here that this net also has six faces. If you were to fold this figure using the line segments, it would create a cube.

To calculate surface area of a cube, find the area of each of the six faces and then add them all to get the surface area.

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

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

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

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

The surface area of the cube is

.Now let’s look at a rectangular prism net.

Start by finding the area of each rectangle.

There are two long faces, two short faces, one bottom, and one top that matches it.

First, let’s find the area of the bottom. It has a length of

and a width of . Since the shape of the bottom is a rectangle, you can use the formula for finding the area of a rectangle.\begin{align*}
\begin{array}{rcl}
A &=& lw \\
A &=& (6)(4) \\
A &=& 24 \ sq. in.
\end{array}\end{align*}

Since the top and bottom match, multiply this area by two:

.Next, find the area of the two long faces. Each long face is a rectangle in shape. The length is

and the width is .\begin{align*}\begin{array}{rcl}
A &=& lw \\
A &=& (6)(2) \\
A &=& 12 \ sq. in.
\end{array}\end{align*}

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

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

Next, find the area of the two shorter faces. Each face is a small rectangle. The length is

and the width is .\begin{align*}\begin{array}{rcl}
A &=& lw \\
A &=& (4)(2) \\
A &=& 8 \ sq. in.
\end{array}\end{align*}

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

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

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

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

The answer is 88 sq. inches.

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.

To find the surface area, you 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*}
\begin{array}{rcl}
A &=& lw \\
A &=& 7(3) \\
A &=& 21 \ sq. cm.
\end{array}\end{align*}

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

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

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

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

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

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

There are two triangles, so multiply this base by two.

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

Now, add all of the areas.

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

The answer is 96 sq. cm.

Let’s look at another example.

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

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

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

You can use this same method with a triangular prism. Let’s look at an example.

Substitute the given values into the formula and solve.

\begin{align*}\begin{array}{rcl}
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{array}\end{align*}

The answer is 104 sq. in.

### Examples

#### Example 1

Earlier, you were given a problem about Leola and the chest she is upholstering.

Leola needs to know the total surface area of the chest so she can order the covering. The net is pictured below.

The chest’s measurements are

long wide by high.To figure out the surface area of the chest, Leola can use the formula for finding the surface area of a rectangular prism.

First, Leola needs to substitute the given information into the formula.

\begin{align*}\begin{array}{rcl}
SA &=& 2(24(18) + 24(18) + (18)18) \\
SA &=& 2(432 + 432 + 324) \\
SA &=& 1188 \ \text{square inches}
\end{array}\end{align*}

Leola will need to purchase 1188 square inches of material.

#### Example 2

Find the surface area of the cube.

To solve this problem, you can use the formula for surface area.

\begin{align*}A = s^2\end{align*}

First, substitute the given value into the formula. The length of one of the sides of the square face is 4 inches.

\begin{align*}\begin{array}{rcl} A &=& 4^2 \\ A &=& 16 \ sq. in. \end{array}\end{align*}

This is the area for one square face. There are six square faces.

Next, multiply 16 sq. in. by six.

The surface area of the cube is

.#### Example 3

Find the surface area of the prism.

To find the surface area of this rectangular prism, use the following formula.

First, substitute the given values into the formula. The length of the prism is 15 inches, the width is 5 inches and the height is 8 inches.

\begin{align*}\begin{array}{rcl} SA &=& 2(15(5) + 15(8) + (5)8) \\ SA &=& 2(75 + 120 + 40) \\ SA &=& 2(235) \\ SA &=& 470 \ sq. in. \end{array}\end{align*}

The answer is 470 sq. inches.

#### Example 4

Find the surface area of the prism.

First, figure out the areas of the bottom, right side, left side and two bases which are triangles.

The bottom is a rectangle. It has a length of 6 m and a width of 4 m.

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

The left side is a rectangle. It has a length of 6 m and a width of 3 m.

\begin{align*} \begin{array}{rcl} A &=& lw \\ A &=& 3(6) \\ A &=& 18 \ sq. m. \end{array}\end{align*}

The right side is a rectangle. It has a length of 6 m and a width of 3 m.

\begin{align*}\begin{array}{rcl} A &=& lw \\ A &=& 6(3) \\ A &=& 18 \ sq. m. \end{array}\end{align*}

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

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

There are two triangles, so multiply this base by two.

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

Next, add all of the areas.

\begin{align*}\begin{array}{rcl} SA &=& 24 + 18 + 18 + 12 \\ SA &=& 72 \ sq. m. \end{array}\end{align*}

The answer is 72 sq. meters.

#### Example 5

Find the surface area of the prism.

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

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

First, substitute the given values into the formula. The length of the prism is 1 cm, the width is 2 cm, and the height is 13 cm.

\begin{align*}\begin{array}{rcl} SA &=& 2(1(2) + 1(13) + (2)13) \\ SA &=& 2(2 +13 + 26) \\ SA &=& 2(41) \\ SA &=& 82 \ sq. cm \end{array}\end{align*}

The answer is 82 sq. cm.

### Review

Use the formula for surface area to find the surface area of each rectangular prism.

- A rectangular prism with a length of 10 in, width of 8 in and height of 5 inches.
- A rectangular prism with a length of 8 in, width of 8 in and height of 7 inches.
- A rectangular prism with a length of 12 m, width of 4 m and height of 6 meters.
- A rectangular prism with a length of 10 in, a width of 6 in and a height of 7 inches.
- A rectangular prism with a length of 12 m, a width of 8 m and a height of 5 meters.
- A rectangular prism with a length of 9 ft, a width of 7 feet and a height of 6 feet.
- A rectangular prism with a length of 10 m, a width of 8 m and a height of 2 m.
- A rectangular prism with a length of 6 ft, a width of 5 feet and a height of 3 feet.

Use the following figure to answer each question.

- What unit are the measurements of this figure?
- What is the length of the base?
- What is the width of the base?
- What is the measure of the triangular side?
- What is the slant height?
- What is the formula for finding the surface area of a triangular prism?
- What is the surface area of this figure?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.14.