## 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.

$\times$ 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. We call this total area the surface area of the figure.

How can we figure out 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. A net is a pattern for a 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 we 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.

Let’s look at an example with a cube.

Example 1

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 $54 \ in^2$.

Example 2 Now let’s look at an example and find the surface area of a rectangular prism using a 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 sides.

There are two short sides.

There is one bottom.

First, we 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.

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

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

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

Since there are two long sides 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 sides. Each side is small rectangle. The length of the short side is 4” and the width is 2”.

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

Since there are two short sides 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 sides.

$SA = 16 + 24 + 24 = 64 \ sq. \ inches$

triangular prism is a prism with two parallel congruent bases just like a rectangular prism. However, a rectangular prism has two parallel congruent bases that are rectangles. Notice the name “rectangle” prism. A triangular prism has 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.

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