10.3: Surface Area of Prisms
Introduction
The Doll House
On her first day wrapping boxes, Candice had a customer come up to the counter with a huge box in her arms.
“This is a doll house for my niece,” the woman said. “Can you please wrap it?”
“Certainly,” Candice said looking at the box.
The box was \begin{align*}40" \ L \times 23" \ W \times 32" \ H\end{align*}
“How big is this roll?” Candice asked Mrs. Scott.
“It is \begin{align*}24" \ W \times 900\end{align*}
“That should be big enough,” Candice thought.
Candice isn’t sure that the roll of paper will be enough for the doll house. This is a problem that can be solved using surface area. If Candice can figure out the surface area of the box, then she will know how much of the wrapping paper she will need to wrap the doll house.
This lesson will teach you all about surface area. Pay attention and at the end of the lesson you will know how to figure out the wrapping paper dilemma.
What You Will Learn
By the end of this lesson, you will have an understanding of the following skills.
- Recognize the surface area of prisms as the sum of the areas of the faces using nets.
- Find surface areas of rectangular prisms using formulas.
- Find surface areas of triangular prisms using formulas.
- Solve real-world problems involving surface area of prisms.
Teaching Time
I. Recognize Surface Area of Prisms as the Sum of the Areas of the Faces using Nets
In this lesson, we will look at prisms with more detail. Remember that a prism is a three-dimensional object with two congruent parallel bases. The shape of the base names the prism and there are rectangles for the sides of the prism.
When we worked with two-dimensional figures, we measured the area of those figures. The area is the space that is contained in a two-dimensional figure. Now we are going to look at the area of three-dimensional figures. Only this isn’t called simply area anymore, it is called surface area.
What is surface area?
The surface area is the covering of a three-dimensional figure. Imagine you could wrap a three dimensional figure 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, we must be able to calculate the area of each face and then add these areas together.
One way to do this is to use a net. Remember that a net is a two-dimensional representation of a three-dimensional solid. A net is a stretched out, or unfolded version of a solid.
If we find the area of each section of the net and then add up those measurements, then we will know the measurement of the “cover” of the figure.
We can do this with prisms of all different kinds. Let’s look at a net for a rectangular prism.
Now we can find the area of each part of the rectangular prism. Notice that the rectangular prism is made up of rectangles. To find the area of a rectangle, we use this formula.
\begin{align*}A=lw\end{align*}
Next we can find the area of each part of the prism. Remember that there are six faces that we need to measure!
\begin{align*}&\text{Bottom face} && \text{top face} && \text{long side} && \text{long side} && \text{short side} && \text{short side}\\
& A = lw && A = lw && A = lh && A = lh && A = wh && A = wh\\
& 12 \times 7 \quad + && 12 \times 7 \quad + && 12 \times 3 \quad + && 12 \times 3 \quad + && 7 \times 3 \quad + && 7 \times 3\\
& 84 \qquad \ \ + && 84 \qquad \ \ + && 36 \qquad \ \ + && 36 \qquad \ \ + && 21 \qquad \ + && 21 \qquad = && \ 282 \ in.^2\end{align*}
The answer is 282 sq. inches.
We found the area of each rectangular face and then added all of these areas together. The total surface area of the rectangular prism is 282 square inches. Using a net helped us to locate all of the faces and find the measurements of each side.
Example
What is the surface area of the figure below?
The first thing we need to do is draw a net. Get ready to exercise your imagination! It may help to color the top and bottom faces to keep you on track. Begin by drawing the bottom face. It is a triangle. Each side of the face is connected to a side face. What shape is each side face? They are rectangles, so we draw rectangles along each side of the triangular base. Lastly, we draw the top face, which can be connected to any of the side faces.
Next let’s fill in the measurements for the sides of each face so that we can calculate their area. Be careful! This time two of the faces are triangles. Remember, we calculate the area of triangles with the formula \begin{align*}A = \frac{1}{2}bh\end{align*}. We need to know the height of the triangles, look at the diagram to find it. We are going to find the areas of two triangles and three rectangles.
Now that we have the measurements of all the faces, let’s calculate the area of each. Remember to use the correct area formula.
\begin{align*}& \text{Bottom face} && \text{Top face} && \text{side} && \text{side} && \text{side} \\ & A = \frac{1}{2}bh && A = \frac{1}{2}bh && A = lw && A = lw && A = lw \\ & \frac{1}{2} (8)(6) \ \ + && \frac{1}{2} (8)(6) \ \ + && 17 \times 10 \ \ + && 17 \times 10 \ \ + && 17 \times 8\\ & 24 \qquad \quad + && 24 \qquad \quad + && 170 \qquad \ + && 170 \qquad \ + && 136 \qquad \ = && \ 524 \ cm^2\end{align*}
We used the formula \begin{align*}A = \frac{1}{2} bh\end{align*} to find the area of the top and bottom faces. We used the formula \begin{align*}A = lw\end{align*} to find the area of the three side faces.
When we add these together, we get a surface area of 524 square centimeters for this triangular prism.
II. Find Surface Area of Rectangular Prisms Using Formulas
Nets let us see each face so that we can calculate the area. However, we can also use a formula to represent the faces as we find their area. Let’s look again at our calculations for the rectangular prism we dealt with.
\begin{align*}& \text{Bottom face} && \text{Top face} && \text{long side} && \text{long side} && \text{short side} && \text{short side}\\ & A = lw && A = lw && A = lh && A = lh && A = wh && A = wh\\ & 12 \times 7 \quad + && 12 \times 7 \quad + && 12 \times 3 \quad + && 12 \times 3 \quad + && 7 \times 3 \quad + && 7 \times 3\\ & 84 \qquad \ \ + && 84 \qquad \ \ + && 36 \qquad \ \ + && 36 \qquad \ \ + && 21 \qquad \ + && 21 \qquad = && \ 282 \ in.^2\end{align*}
Notice that our calculations repeat in pairs. This is because every face in a rectangular prism is opposite a face that is congruent. In other words, the top and bottom faces have the same measurements, the two long side faces have the same measurements, and the two short side faces have the same measurements. Therefore we can create a formula for surface area that gives us a short cut. We simply double each calculation to represent a pair of faces.
The formula looks like this.
\begin{align*}SA = 2lw + 2lh + 2hw\end{align*}
In this rectangular prism, \begin{align*}l = 12\end{align*} inches, \begin{align*}w = 7\end{align*} inches, and \begin{align*}h = 3\end{align*} inches. We simply put these numbers into the formula and solve for surface area. Let’s try it.
\begin{align*}SA & = 2lw + 2lh + 2hw\\ SA & = 2(12 \times 7) + 2 (12 \times 3) + 2(3 \times 7)\\ SA & = 2(84) + 2(36) + 2(21)\\ SA & = 168 + 72 + 42\\ SA & = 282 \ in.^2\end{align*}
As we have already seen, the surface area of this prism is 282 square inches. This formula just saves us a little time by allowing us to calculate the area of pairs of faces at a time. Let’s try another example.
Example
What is the surface area of the figure below?
All of the faces of this prism are rectangles, so we can use our formula for finding the surface area of rectangular prisms. We simply put the measurements into the formula and solve for \begin{align*}SA\end{align*}, surface area.
\begin{align*}SA & = 2lw + 2lh + 2hw\\ SA & = 2(21 \times 14) + 2 (21 \times 5) + 2(5 \times 14)\\ SA & = 2(294) + 2(105) + 2(70)\\ SA & = 588 + 210 + 140\\ SA & = 938 \ cm^2\end{align*}
This rectangular prism has a surface area of 938 square centimeters. If you’re not sure which measurements go with which side of the prism, try drawing a net.
Let’s practice using this formula.
10D. Lesson Exercises
Use the formula to find the surface area of each rectangular prism.
- Length of 8 in, width of 4 inches, height of 6 inches
- Length of 5 ft, width of 4 ft, height of 2 ft
Take a few minutes to check your answers.
Write this formula for finding the surface area of a rectangular prism down in your notebook.
Now let’s see how we can use a formula to find the surface area of a triangular prism.
III. Find Surface Areas of Triangular Prisms using Formulas
Triangular prisms have a different formula for finding surface area because they have two triangular faces opposite each other. Remember, the formula for the area of triangles is not the same as the area formula for rectangles, so we’ll have to proceed differently here to find a formula for surface area.
First, we know that we need to find the area of the two triangular faces. Each face will have an area of \begin{align*}\frac{1}{2} bh\end{align*}. Remember, we can use a formula to calculate the area of a pair of faces. Therefore we can double this formula to find the area of both triangular faces at once. This gives us \begin{align*}2(\frac{1}{2} bh)\end{align*}. The numbers cancel each other out, and we’re left with \begin{align*}bh\end{align*}. That part was easy!
Next, we need to calculate the area of each side face. The length of each rectangle is the same as the height of the prism, so we’ll call this \begin{align*}H\end{align*}. The width of each rectangle is actually the same as each side of the triangular base. Instead of multiplying the length and width for each rectangle, we can combine this information. We can multiply the perimeter of the triangular base, since it is the sum of each “width” of a rectangular side, by the height of the prism, \begin{align*}H\end{align*}.
If we put these pieces together—the area of the bases and the area of the side faces—we get this formula.
\begin{align*}SA = bh + (s1 + s2 + s3)H\end{align*}
To use this formula, we fill in the base and height of the prism’s triangular base, the lengths of the base’s sides, and the height of the prism. Don’t confuse the height of the triangular base with the height of the prism! Let’s try out the formula.
Example
What is the surface area of the figure below?
We have all of the measurements we need. Let’s put them into the formula and solve for surface area, \begin{align*}SA\end{align*}.
\begin{align*}SA & = bh + (s1 + s2 + s3)H\\ SA & = 4(3) + (5 + 5 + 4) (15)\\ SA & = 12 + 14 (15)\\ SA & = 12 + 210\\ SA & = 222 \ cm^2\end{align*}
This triangular prism has a surface area of 222 square centimeters.
The biggest thing that we need to watch out for with this formula is that we put the correct measurement in the correct spot.
IV. Solve Real-World Problems Involving Surface Area of Prisms
We have learned two ways to find surface area: drawing a net or using a formula.
Write both of these formulas down in your notebooks. Then continue.
We can use either of these methods to solve word problems involving surface area. Nets may be especially useful if the problem does not provide an image of the prism. If you choose to use a formula, be sure you know whether the problem deals with a rectangular or a triangular prism. Let’s practice using what we have learned.
Example
Crystal is wrapping the box below in wrapping paper for her brother’s birthday. How much wrapping paper will she need?
First of all, is this a rectangular or triangular prism? All of the faces are rectangles, so it is a rectangular prism. The picture clearly shows us what its length, width, and height are, so let’s use the formula for finding the surface area of rectangular prisms. Simply put the measurements in for the appropriate variables in the formula.
\begin{align*}SA & = 2lw + 2lh + 2hw\\ SA & = 2(12 \times 9) + 2 (12 \times 6) + 2(6 \times 9)\\ SA & = 2(108) + 2(72) + 2(54)\\ SA & = 216 + 144 + 108\\ SA & = 468 \ in.^2\end{align*}
Crystal will need 468 square inches of wrapping paper in order to cover the present.
Now let’s go back to the problem from the introduction and use what we have learned to figure out the surface area.
Real–Life Example Completed
The Doll House
Here is the original problem once again. Reread the problem and underline the important information.
On her first day wrapping boxes, Candice had a customer come up to the counter with a huge box in her arms.
“This is a doll house for my niece,” the woman said. “Can you please wrap it?”
“Certainly,” Candice said looking at the box.
The box was \begin{align*}40" L \times 23" W \times 32" H\end{align*}. Candice looked at the different rolls of wrapping paper. She was sure that she was going to need the largest roll of wrapping paper that she could find, and even then she was sure that it was going to take two pieces of paper. The wrapping station has a huge roll of paper on a big roller. This one was brand new.
“How big is this roll?” Candice asked Mrs. Scott.
“It is \begin{align*}24" W \times 900\end{align*} feet long," Mrs. Scott said.
“That should be big enough,” Candice thought.
To figure out if the wrapping paper on the roll will be enough to wrap the doll house, Candice has to figure out the surface area of the box. She knows the dimensions, so she can use these dimensions to help solve the problem.
The box is \begin{align*}40"L \times 23"W \times 32" H\end{align*}.
Since a box is a rectangular prism, and we know the length and width, we can use this formula for the surface area of the box.
\begin{align*}SA=2(lw+lh+wh) \end{align*}
Next, we can substitute the given measurements into the formula.
\begin{align*}SA & =2 \cdot [40(23)+ 40(32)+23(32)] \\ SA & =2(920+1280+736) \\ SA & =2(2936) \\ SA & =5872 \ sq.inches\end{align*}
Let’s change that into square feet by dividing by 144.
\begin{align*}5872 \div 144 = 40.7778 \ sq. feet\end{align*}
We can round up to 41 square feet just to be sure.
Now the measurement of the wrapping paper was in inches and feet.
\begin{align*}24" \times 900 \ ft\end{align*}, so let’s change it to feet
\begin{align*}2 \ ft \times 900 \ ft = 1800 \ sq. feet\end{align*}
The wrapping paper will be enough to cover the box of the doll house.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Prism
- a three-dimensional solid with two congruent parallel bases.
- Area
- the space enclosed inside a two-dimensional figure.
- Surface Area
- the covering of a three dimensional solid.
- Net
- a two-dimensional representation of a three-dimensional solid.
- Rectangular Prism
- a prism with rectangles as bases and faces.
- Triangular Prism
- a prism with triangles as bases and rectangles as faces.
Technology Integration
- http://www.mathplayground.com/mv_surface_area_prisms.html – This is a Brightstorm video on how to find the surface area of a prism.
Time to 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.
9. A rectangular prism with a length of 3 feet, a width of 6 feet and a height of 2 feet.
10. A rectangular prism with a length of 4 feet, a width of 4 feet and a height of 4 feet.
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