Surface Area of Right Prisms
14.1: Multifaceted (5 minutes)
The purpose of this warm-up is for students to recognize important parts of solids in anticipation of computing volume and surface area. The figure used in the next activity is introduced in this warm-up as a way for students to start thinking about parts of solids and how we use them to compute surface area or volume.
Arrange students in groups of 2. Display the prism assembled from the blackline master for all to see. Give students 1 minute of quiet think time followed by time to discuss their ideas with a partner. Follow with a whole-class discussion.
Your teacher will show you a prism.
- What are some things you could measure about the object?
What units would you use for these measurements?
Select students to share their responses. Ask students to think about units that do not make sense to use for measurements (feet, miles, yards, etc). Invite students to share their explanations of why these units do not make sense to use.
14.2: So Many Faces (15 minutes)
In this activity, students make sense of three different methods for calculating the surface area of a figure. Three different methods are described to students and they are asked to determine which one they agree with (if any) (MP3). They then think about generalizing the methods to figure out if they would work for any prism. This activity connects to work they did with nets in a previous grade and builds upon strategies students might have to calculate surface area.
As students work on the task, monitor for students who understand the different methods and can explain if any of them will work for any other prisms.
Note: It is not important for students to learn the term “lateral area.”
Arrange students in groups of 2. Display the prism assembled previously in the warm-up for all to see. Ask students: “how might we find surface area of this prism?” Invite students to share their ideas. Give students 1 minute of quiet think to read Noah’s method for calculating surface area followed by time discuss whether they agree with Noah or not. Repeat this process for the remaining two methods. Once all three methods have been discussed give students 1–2 minutes of quiet work time to answer the rest of the questions in the task statement.
Supports accessibility for: Visual-spatial processing
Here is a picture of your teacher's prism:
Three students are trying to calculate the surface area of this prism.
- Noah says, “This is going to be a lot of work. We have to find the areas of 14 different faces and add them up.”
- Elena says, “It’s not so bad. All 12 rectangles are identical copies, so we can find the area for one of them, multiply that by 12 and then add on the areas of the 2 bases.”
- Andre says, “Wait, I see another way! Imagine unfolding the prism into a net. We can use 1 large rectangle instead of 12 smaller ones.”
Do you agree with any of them? Explain your reasoning.
- How big is the “1 large rectangle” Andre is talking about? Explain or show your reasoning. If you get stuck, consider drawing a net for the prism.
- Will Noah’s method always work for finding the surface area of any prism? Elena’s method? Andre’s method? Be prepared to explain your reasoning.
Which method do you prefer? Why?
Students may think that Andre’s method will not work for all prisms, because it will not work for solids that have a hole in their base and therefore more lateral area on the inside. Technically, these solids are not prisms, because their base is not a polygon. However, students could adapt Andre’s method to find the surface area of a solid composed of a prism and a hole.
Select previously identified students to share their reasoning. If not brought up in students' explanations, display the image for all to see and point out to students that the length of the “1 big rectangle” is equal to the perimeter of the base.
Students may have trouble generalizing which method would work for any prism. Here are some guiding questions:
- “Which of the students’ methods will work for finding the surface area of this particular prism?” (all 3)
- “Which of the students’ methods will work for finding the surface area of any prism?” (Noah’s and Andre’s)
- “Which of the students’ methods will work for finding the surface area of other three-dimensional figures that are not prisms?” (only Noah’s)
If not mentioned by students, be sure students understand:
- Noah’s method will always work, but it can be inefficient if there are a lot of faces.
- Elena’s method will not always work because the rectangles will not always be the same size, but we can notice that some shapes are the same and not have to work them all out individually.
- Andre’s method does always work even if the rectangles have different widths. The length of the rectangle will be the same as the perimeter of the base and the width of the rectangle will be the height of the prism.
- Prisms can always be cut into three pieces: two bases and one rectangle whose length is the perimeter of a base and whose width is the height of the prism. This can be more efficient than the other methods because students only need to calculate two areas (since the two bases will be identical copies).
- This method only works for prisms. For other shapes, such as pyramids, Noah’s method of finding all the faces individually or Elena’s method of combining those faces into identical copy groups will work. Solids with holes, such as the triangular prism with a square hole, can use a variation on Elena’s method: two congruent triangles with holes for the bases, one rectangle for the outside side faces, and another rectangle for the faces forming the hole.
Explain to students that they will have the opportunity in the next activity to practice using any of these strategies.
Provide students with prompts for feedback that will help their partners strengthen their ideas and clarify their language (e.g., “Can you draw a picture to support your explanation?”, “You should expand on….”, “How does that match with Andre’s thinking?”, etc.). Invite students to go back and revise or refine their written explanation based on their peer feedback. These conversations will help students make sense of the different methods for calculating the surface area of a figure.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)
14.3: Revisiting a Pentagonal Prism (15 minutes)
In this activity, students are presented with a figure that was used in a previous lesson to explore volume. Here, they explore its surface area and compare different methods from the previous task. Students work with a partner to share the task of investigating two methods to calculate the surface area.
As students work on the task, listen for students who find similarities and differences between the method they used and the one their partner used.
Arrange students in groups of 2. Tell students that they might recognize this figure from a previous lesson, but today they are going to compare two different methods for calculating its surface area.Give students 1–2 minutes of quiet work time followed by time to trade their work with a partner to compare answers and methods. Follow with a whole-class discussion.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Between you and your partner, choose who will use each of these two methods to find the surface area of the prism.
- Adding the areas of all the faces
- Using the perimeter of the base.
Use your chosen method to calculate the surface area of the prism. Show your thinking. Organize it so it can be followed by others.
- Trade papers with your partner, and check their work. Discuss your thinking. If you disagree, work to reach an agreement.
Are you ready for more?
In a deck of cards, each card measures 6 cm by 9 cm.
When stacked, the deck is 2 cm tall, as shown in the first photo. Find the volume of this deck of cards.
Then the cards are fanned out, as shown in the second picture. The distance from the rightmost point on the bottom card to the rightmost point on the top card is now 7 cm instead of 2 cm. Find the volume of the new stack.
Select previously identified students to share the discussion they had with their partner. To highlight the difference between the two methods, ask:
- “How did you find the area of the base?”
- “How did you find any other areas you needed to solve the problem?”
- “How many different shapes did you need to calculate the area of when using the first method (calculating area of all the faces)?”
- “How many different shapes did you need to calculate the area of when using the second method (using perimeter of base)?”
- “Which method do you prefer for this problem? Why?”
- “Do you think you will prefer the same method for every problem? Why or why not?”
- “What would make you change methods?”
- “Do you need to know all of the measurements in the picture to solve for surface area?” (No, you just need to know the perimeter and area of the base and the height of the figure.)
- “Could you solve for volume with the measurements given in the picture? If so, are there any unnecessary measurements? If not, what else would you need to know?”
If not brought up in students’ explanations, explain to students that the first method requires finding the area of 6 different shapes (there are 7 faces, but the two bases are the same). While the calculations using this method were simple, there were more pieces. The second method requires visualizing the solid in a different way, but we only needed to find the area of two different pieces (the long rectangle and base).
- “What is surface area?” (The total area of all the exposed faces of an object.)
- “What are some methods for calculating surface area of prisms?” (Find the area of each face and add them for the total. Find groups of faces that have the same area and save some computation. Find the area of the bases and add that to the area of a “long rectangle.”)
14.4: Cool-down - Surface Area of a Hexagonal Prism (5 minutes)
Student Lesson Summary
To find the surface area of a three-dimensional figure whose faces are made up of polygons, we can find the area of each face, and add them up!
Sometimes there are ways to simplify our work. For example, all of the faces of a cube with side length \(s\) are the same. We can find the area of one face, and multiply by 6. Since the area of one face of a cube is \(s^2\), the surface area of a cube is \(6s^2\).
We can use this technique to make it faster to find the surface area of any figure that has faces that are the same.
For prisms, there is another way. We can treat the prism as having three parts: two identical bases, and one long rectangle that has been taped along the edges of the bases. The rectangle has the same height as the prism, and its width is the perimeter of the base. To find the surface area, add the area of this rectangle to the areas of the two bases.