A cup of paint covers about 22 square feet. You need to paint all faces of a cube to use as a prop in a play. Each edge of the cube is 2.5 feet long. How much paint will you need to buy?

### Surface Area and Nets

A **net** is a drawing that shows the edges and faces of a solid in two dimensions. You can think of a net as what you would get if you “unfolded” a solid. In the series of images below, you can see a triangular prism, the triangular prism starting to be unfolded, and the net for the triangular prism.

The **surface** **area** of a solid is the sum of the area of all its faces. This means that one way to find the surface area of a solid is to find the area of its net.

#### Approximating the Surface Area

Find the approximate surface area of the triangular prism below. The base is an equilateral triangle.

The net is made of three congruent rectangles and two congruent equilateral triangles.

The surface area is the sum of the areas of the five shapes. To find the area of the triangle, you need to know the height of the triangle. Using the Pythagorean Theorem, you can determine that the height is approximately 3.46 inches.

\begin{align*}Area_{Triangle} &=\frac{bh}{2}=\frac{4\cdot 3.46}{2}=6.93 \ in^2 \\
Area_{Rectangle} &=bh=4 \cdot 6=24 \ in^2 \\
Total \ Surface \ Area &=2(6.93)+3(24)=85.86 \ in^2\end{align*}

Draw the net of a square pyramid.

A square pyramid has a height of 20 inches. Each side of the square base is 12 inches. What is the surface area of the pyramid?

In order to find the surface area, you will need to determine the area of the triangle faces. In order to find the area of each triangle face, you will need the base and height of the triangle. In order to do this, imagine a right triangle standing upright in the pyramid.

The height of this triangle is 20 inches and the base of the triangle is 6 inches. It's hypotenuse, which is the height of the triangle face, can be determined with the Pythagorean Theorem.

\begin{align*}6^2+20^2=c^2\rightarrow c\approx 20.88 \ in\end{align*}

Now you can find the area of each of the five shapes that make up the net in order to find the surface area.

\begin{align*}Area_{Triangle} &=\frac{bh}{2}=\frac{(12\cdot 20.88)}{2}=125.28 \ in^2 \\
Area_{Square} &=12^2=144 \ in^2\\
Total \ Surface \ Area &=4(125.28)+144=645.12 \ in^2
\end{align*}

### Examples

#### Example 1

Earlier, you were asked how much paint will you need to buy.

One cup of paint covers about 22 square feet. Each edge of the cube is 2.5 feet long. In order to figure out how much paint you will need, you should find the surface area of the cube.

The cube has six congruent square faces. The area of each square is \begin{align*}(2.5)^2=6.25 \ ft^2\end{align*}. The total surface area of the cube is \begin{align*}37.5 \ ft^2\end{align*}. You will need \begin{align*}\frac{37.5}{22}\approx 1.7\end{align*} cups of paint.

#### Example 2

Draw the net for a pentagonal prism.

#### Example 3

The base of a pentagonal prism is the following pentagon:

Find the area of the pentagon by dissecting the pentagon into five triangles and finding the area of each triangle.

Use the Pythagorean Theorem to find the height of each triangle.

\begin{align*}5^2+b^2=8.5^2\rightarrow b \approx 6.87\end{align*}

The area of each triangle is approximately \begin{align*}\frac{bh}{2}=\frac{10 \cdot 6.87}{2}=34.37 \ in^2\end{align*}. Therefore, the area of the pentagon is \begin{align*}34.37 \cdot 5=171.85 \ in^2\end{align*}.

#### Example 4

Find the surface area for a pentagonal prism with a height of 25 inches and a base.

The pentagonal prism is made of five rectangular faces and two pentagonal faces. Each rectangle is 10 inches by 25 inches.

\begin{align*}Area_{Pentagon} &=171.85 \ in^2 \\ Area_{Rectangle} &=(10 \cdot 25)=250 \ in^2\\ Total \ Surface \ Area &=2(171.85)+5(250)=1593.69 \ in^2 \end{align*}

### Review

1. Explain the connection between the surface area of a solid and the net of a solid.

2. When stating a surface area, why do you use square units such as “\begin{align*}in^2\end{align*}”?

A triangular pyramid has four congruent equilateral triangle faces. Each edge of the pyramid is 6 inches.

3. Draw a net for the pyramid.

4. Find the area of one triangle face.

5. Find the surface area of the pyramid.

A 20 inch tall hexagonal pyramid has a regular hexagon base that can be divided into six equilateral triangles with side lengths of 12 inches, as shown below.

6. Draw a net for the pyramid.

7. Find the area of the hexagon base.

8. The pyramid has 6 triangular faces. Use the Pythagorean Theorem to help you to find the height of each of these triangles.

9. Find the total surface area of the pyramid.

A square prism is topped with a square pyramid to create the composite solid below.

10. Draw a net for the solid.

11. There are four triangular faces. Use the Pythagorean Theorem to help you find the height of each of these triangles.

12. Find the total surface area of the solid.

A solid has the following net.

13. What type of solid is this?

14. Find the surface area of the solid.

15. What would the net of a cylinder look like? Try to make a sketch.

16. What would the net of a cone look like? Try to make a sketch.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.11.