# Lesson 16

Surface Area and Volume

## 16.1: Maximize Area (5 minutes)

### Warm-up

The purpose of this warm-up is for students to reason about how to manipulate a shape to enclose more space using a limited amount of material. In subsequent activities, students will apply this same kind of thinking to three-dimensional shapes.

### Launch

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

### Student Facing

The zoo wants to give the elephants as much space as possible in a rectangular enclosure meant for feeding. The zoo has 180 feet of fencing. What should the dimensions of the rectangle be? Be prepared to share your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may not be certain if a square is a rectangle. Remind them that a rectangle is defined as a quadrilateral with 4 right angles, and ask them if a square meets that definition.

### Activity Synthesis

The purpose for discussion is for students to understand that the square has the most area for a fixed perimeter. Select students to share their responses and explain why they think their enclosure has the most area. Consider these questions for discussion:

• “What are two opposite extreme cases that have very small area? What would be the medium between these two extremes?” (Two extreme cases would be if the fence was 1 foot wide and 89 feet across or 1 foot across and 89 feet wide. The medium between these two extremes would be a square enclosure, which would be 45 feet by 45 feet.)
• “What if we wanted to maximize the volume of a prism with fixed surface area instead of maximizing area of a rectangle with fixed perimeter?” (It seems likely that a cube would give us the maximum volume of a rectangular prism with fixed surface area.)

## 16.2: Maximize in Three Dimensions (15 minutes)

### Activity

Students analyze the relationship between the dimensions and the surface area of a solid with fixed volume. As students identify patterns in the results of their classmates’ calculations, they are making sense of the problem and persevering to solve it (MP1).

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

This activity involves thinking about the workings of a lithium ion battery, such as those in laptops or cell phones. Emphasize to students that under no circumstances should they attempt to open one of these batteries, as this is extremely dangerous.

Arrange students in groups of 4.

Prepare a 5-column table in which students can record the results of their calculations for all to see. It will be easiest to organize the data by putting the numbers in order (small, medium, large), rather than arbitrarily assigning direction (length, width, height). To illustrate this idea, invite students to calculate the volume of these rectangular prisms:

1. length = 2, width = 6, height = 7
2. length = 6, width = 7, height = 2

Point out that the assignment of the dimensions didn’t matter; the two prisms had the same volume. Tell students to record their data in the class chart by numerical order rather than in order of length, width, and height.

Design Principle(s): Support sense-making
Representation: Internalize Comprehension. Activate or supply background knowledge about the surface area of a rectangular prism. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

1. Find a set of dimensions for a rectangular prism with volume 60 cubic units. Calculate the surface area of your prism. Add your data to the class chart.
2. A lithium ion battery contains a rectangular prism made of lithium. The energy in the battery is proportional to the surface area of this prism. Assume the lithium has a fixed volume of 60 cubic millimeters. Find the dimensions of a rectangular prism with this volume that maximizes its surface area. What is its surface area?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Minimizing surface area plays a large role in manufacturing. Companies try to use the smallest amount of resources possible to package products in order to save money.

1. Calculate the surface area of the 2 figures shown, which both enclose the same volume.
2. Which container would you recommend a company use to package small candies? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may assume that because a square maximizes area for a set perimeter, a cube must maximize surface area for a set volume. It’s true that a cube maximizes volume for a set surface area, but not the other way around. Prompt these students to consider the results they are seeing from their classmates’ calculations to verify if this assumption holds true.

If students struggle to calculate the surface area of their prism, remind them that they can draw the faces of the prism to help them organize their thinking.

### Activity Synthesis

Sort the data students have collected in descending numerical order of the surface areas. Consider using a spreadsheet to organize the data and displaying it for all to see. Ask students, “What if we considered values that aren’t integers?” The important takeaway is that to maximize surface area for a set volume, a shape should be made flat and long. For a battery, this is accomplished by making the lithium into a thin foil that can be rolled up to fit inside the battery.

Consider discussing the general idea that packing as much surface area as possible into a small volume is often accomplished by folding. For example, the chemical reactions that occur in mitochondria to provide energy to cells in our bodies take place on the surface, which is why the inside of a mitochondria has many folds. Likewise, the chemical reactions that occur in the brain happen on its surface, which is why the brain has many folds. Intestines absorb nutrients through their surfaces, so they are folded to pack as much surface area into the digestive tract as possible.

## 16.3: Assume a Spherical Elephant (10 minutes)

### Activity

In this activity, students calculate the surface area to volume ratios of animals with different shapes and think about how this ratio might be important in biology. Note that this is the first time students will be using formulas for the surface area and volume of a sphere, and that these formulas are given without derivation. Students used the sphere volume formula in grade 8.

### Launch

“Assume a spherical cow” is a common joke about physics. Sometimes in physics, everything about the world is made as simple as possible. There is no air resistance, and everything is modeled with ideal shapes like a sphere or a point. Tell students they will do the same today with spherical elephants and cylindrical snakes.

### Student Facing

For a sphere with radius $$r$$, its volume is $$\frac43 \pi r^3$$ and its surface area is $$4\pi r^2$$.

1. Let’s model an elephant with a sphere that has a radius of 4.5 feet.
1. What is the volume of the elephant?
2. What is the surface area of the elephant?
2. Let’s model a snake with a cylinder of length 3 feet and diameter 0.2 feet.
1. What is the volume of the snake?
2. What is the surface area of the snake?
3. Compute the surface area to volume ratio, or $$\frac{SA}{V}$$, for each animal.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students how the surface area to volume ratio could affect the biology of snakes and elephants. The goal is for students to notice that the snake has a much higher ratio and to think about the impacts this may have.

One aspect of the biology is that elephants are warm-blooded, so to maintain a consistent body temperature, their bodies need to minimize the surface area through which heat enters or escapes their bodies. Snakes are cold-blooded, so their bodies maximize surface area in order to absorb heat from the environment through their skin.

## 16.4: Measuring Strength (10 minutes)

### Optional activity

Students investigate another application of surface area and volume: measuring strength based on cross-sectional muscle area.

### Launch

Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams of spheres with radius $$\frac{1}{200}$$ unit, 1 unit, and 5 units. If students are unsure where to begin, suggest that they draw and label the radius of each sphere to help organize the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing

### Student Facing

Suppose a human is a sphere with a radius of 1 unit, an ant is a sphere with a radius of $$\frac{1}{200}$$ unit, and an elephant is a sphere with a radius of 5 units.

1. The raw strength of a living creature is the cross-sectional area of its muscles. The cross section of each of our spherical beings is a circle of radius $$r$$ where $$r$$ is the creature’s radius. Order the human, ant, and elephant by their raw strength from least to greatest. Show your reasoning.
2. Relative strength is given by the ratio of raw strength to volume. It measures how strong a creature is for its size. Create an expression for the relative strength of a spherical being with radius $$r$$. (Remember that the raw strength formula is $$\pi r^2$$ and the volume formula for a sphere is $$\frac{4}{3}\pi r^3$$.)
3. Order the human, the ant, and the elephant by their relative strength. Which is the strongest for its size?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Tell students to imagine the heaviest weight an average person can lift. For ease of calculations, consider using a number like 100 pounds. Now ask, “If a human had the relative strength of an ant, how much weight could they lift?” (An ant has 200 times the relative strength of a human. So, if a human was as strong as an ant, they could lift 20,000 pounds. That is the weight of around 5 average-sized cars.)

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their solutions to the questions about relative strength, present an incorrect answer and explanation. For example, “The relative strength is the ratio of raw strength to volume, so the expression for relative strength is the product of raw strength and volume, which is $$\pi r^2 \boldcdot \frac{4}{3} \pi r^3$$.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify the meaning of ratio in this context. Invite students to share their critiques and corrected explanations with the class. This will help students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students computed surface area and volume in various contexts. Ask students what other situations might involve surface area to volume ratios. To prompt their thinking, ask them to consider processes that happen on the surface of an object versus processes that happen inside an object. Examples include:

• Ice cubes of different shapes melt at different rates because melting happens on the surface of an ice cube. A spherical ice cube has the lowest surface area to volume ratio, so spherical ice is ideal if you don’t want it to melt very fast.
• Our lungs have many folds to increase the surface area for oxygen exchange.
• Plants lose water vapor through their surfaces. Plants that live in hot, dry deserts have small leaves to minimize water loss. Cacti take this to an extreme—their “leaves” are spines through which almost no water is lost.

## 16.5: Cool-down - Minimize Area (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

The three prisms shown each have a volume of 216 cubic centimeters. Which prism do you think has the largest surface area? Which do you think has the smallest surface area?

The surface area of the first prism is 492 square centimeters, the surface area of the second is 246 square centimeters, and the surface area of the third is 216 square centimeters. The cube, then, has the smallest surface area. In general, the cube is the rectangular prism with the least amount of surface area for its volume. The 1 by 12 by 18 cm prism has the greatest surface area of the three. If we want a rectangular prism to have more surface area, the best design is to make it wide and long.

Shapes that are more compact, like a cube, have the least surface area for a given volume. But it turns out spheres do even better than cubes. A sphere with radius 3.72 centimeters, shown in the figure, has an approximate volume of 216 cubic centimeters like the earlier prisms, but its approximate surface area is just 174 square centimeters.

We can see examples of maximizing or minimizing surface area in nature. For instance, a snake is cold-blooded, meaning that it gets its heat from the environment. Its long, narrow shape helps it soak up more heat from the sun. On the other hand, large mammals such as elephants and cows are warm-blooded, which means that they produce their own heat internally. Their shapes are more compact, closer to spheres, and this allows them to lose as little heat as possible through their skin.