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# Surface Area of Prisms

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# Surface Area of Prisms

Remember Jillian and her jewelry box? Take a look at this dilemma.

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 $\times$ 6” wide by 4” high. Jillian isn’t sure which measurements go where.

This is where you come in. In this Concept, you will learn how to use a net and a formula to figure out the surface area of prisms.

### Guidance

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

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 $\times$ 3 $=$ 24 sq. in

The area of the rectangle is 24 sq. in or $in^2$ .

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 Concept 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 with a cube first to get the idea.

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.

$A & = s^2\\A & = 3^2\\A & = 9 \ sq. \ in.$

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.

$9(6) & = 54 \ sq. \ in.\\9 + 9 + 9 + 9 + 9 + 9 & = 54 \ in^2$

The surface area of the cube is $54 \ in^2$ .

Now let’s look at 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.

$A & = lw\\A & = (6)(4)\\A & = 24 \ sq. \ in.$

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”.

$A & = lw\\A & = (6)(2)\\A & = 12 \ sq. \ in.$

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

$A = 24 \ sq. \ in$

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

$A & = lw\\A & = (4)(2)\\A & = 8 \ sq. \ in.$

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

$A & = 2(8)\\A & = 16 \ sq. \ in$

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

$SA = 16 + 24 + 48 = 88 \ sq. \ inches$

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.

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.

$A & = lw\\A & = 7(3)\\A & = 21 \ sq. \ cm.$

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

$A & = lw\\A & = 7(4)\\A & = 28 \ sq. \ cm.$

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

$A & = lw\\A & = 7(5)\\A & = 35 \ sq. \ cm.$

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

$A & = \frac{1}{2}bh\\A & = \frac{1}{2}(3)(4)\\A & = \frac{1}{2}(12)\\A & = 6 \ sq. \ cm$

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

$A & = 2(6)\\A & = 12 \ sq. \ cm.$

Now we add up all of the areas.

$SA & = 12 + 35 + 28 + 21\\SA & = 96 \ sq. \ cm.$

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.

$SA = 2(lw + lh + wh)$

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.

$SA & = 2(9(3) + 9(5)+ (3)5)\\SA & = 2(27 + 45 + 15)\\SA & = 2(87)\\SA & = 174 \ sq. \ in.$

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.

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

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

#### Example A

Solution: 235 sq. inches

#### Example B

Solution: 72 sq. meters

#### Example C

Solution: 41 sq. cm.

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 $\times$ 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.

$SA & = 2(lw + lh + wh)\\SA & = 2(7(6) + 7(4) + (4)6)\\SA & = 2(42 + 28 + 24)\\SA & = 188 \ square \ inches$

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.

$12'' \times 12'' & = 144 \ sq. \ in && Nope-too \ small\\13'' \times 13'' & = 169 \ sq. \ in && Nope-still \ too \ small\\14'' \times 14'' & = 196 \ sq. \ in$

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

### Guided Practice

Here is one for you to try on your own.

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

### Vocabulary

Here are the vocabulary words in this Concept.

Surface Area
the total area of all of the surfaces of a solid
Solid Figure
a three-dimensional figure with height, width and depth.
Prism
a solid with two parallel congruent bases.
Triangular Prism
a prism where the faces are triangles
Rectangular Prism
a prism where the faces are prisms
Face
any flat surface on a solid figure
Edge
when two faces meet in a line segment. The line segment is the edge.
Vertex
when three or more faces meet at a single point.

### Practice

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

1. A rectangular prism with a length of 10 in, width of 8 in and height of 5 inches.

2. A rectangular prism with a length of 8 in, width of 8 in and height of 7 inches.

3. A rectangular prism with a length of 12 m, width of 4 m and height of 6 meters.

4. A rectangular prism with a length of 10 in, a width of 6 in and a height of 7 inches.

5. A rectangular prism with a length of 12 m, a width of 8 m and a height of 5 meters.

6. A rectangular prism with a length of 9 ft, a width of 7 feet and a height of 6 feet.

7. A rectangular prism with a length of 10 m, a width of 8 m and a height of 2 m.

8. A rectangular prism with a length of 6 ft, a width of 5 feet and a height of 3 feet.

Directions: Use the following figure to answer each question.

9. What unit are the measurements of this figure?

10. What is the length of the base?

11. What is the width of the base?

12. What is the measure of the triangular side?

13. What is the slant height?

14. What is the formula for finding the surface area of a triangular prism?

15. What is the surface area of this figure?

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