14.1: A Cylinder of Unknown Height
Warm-up: 5 minutes
The purpose of this warm-up is to assess students’ understanding of the volume of a cylinder. Students learned that the volume of either a cylinder or prism is found by multiplying the area of the base by its height. In this warm-up, students are given information to find the area of a cylinder’s base, but they are not given the height. Students propose a volume for the cylinder and explain why it works. Since the diameter of the base is 8, the area of the base is $16\pi$.
If students have trouble getting started, ask them:
- “Do you have enough information to calculate the area of the base?”
- “What is the radius?”
Identify students who use these strategies:
- find the area of the base first then set up the equation $V=16\pi h$.
- choose a specific value for $h$ then solve for the volume.
Arrange students in groups of 2. Tell students that in a previous lesson, they learned how to find the volume of a cylinder if they know the cylinder’s radius and height. Draw their attention to where volume formulas are displayed in the classroom as the unit progresses. Give students 1–2 minutes of quiet work time followed by time to explain their reasoning to their partner. Follow this with a whole-class discussion.
What is a possible volume for this cylinder if the diameter is 8 cm? Explain your reasoning.
The goal of this discussion is for students to communicate how the height of a cylinder is related to its volume. Invite students to share their solutions and their reasoning. Record and display the dimensions and volumes of cylinders that correspond to solutions given by students.
14.2: What’s the Dimension?
Activity: 15 minutes
In this activity, students find the missing dimensions of cylinders when given the volume and the other dimension. A volume equation representing the cylinder is given for each problem.
Identify students who use these strategies: guess and check, divide each side of the equation by the same value to solve for missing variable, or use the structure of the volume equation to reason about the missing variable
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by time to share their explanation for the first problem with their partners.
The volume $V$ of a cylinder with radius $r$ is given by the formula $V=\pi r^2h$.
- The volume of this cylinder with radius 5 units is $50\pi$ cubic units. This statement is true:
$$50\pi = 5^2 \pi h$$
What does the height of this cylinder have to be? Explain how you know.
- The volume of this cylinder with height 4 units is $36\pi$ cubic units. This statement is true:
$$36\pi = r^2 \pi 4$$
What does the radius of this cylinder have to be? Explain how you know.
Select previously identified students to explain the strategies they used to find the missing dimension in each problem. If not brought up in students’ explanations. Discuss the following strategies and explanations:
- Guess and check: plug in numbers for $h$, a value that make the statements true. Since the solutions for these problems are small whole numbers, this strategy works well. In other situations, this strategy may be less efficient.
- Divide each side of the equation by the same value to solve for the missing variable: for example, divide each side of $36\pi=r^2 \pi 4$ by the common factor, $4\pi$. It’s important to remember $\pi$ is a number that can be multiplied and divided like any other factor.
- Use the structure of the equation to reason about the missing variable: for example, $50\pi$ is double $25\pi$, so the missing value must be 2.
14.3: Cylinders with Unknown Dimensions
Activity: 15 minutes
The purpose of this activity is for students to use the structure of the volume formula for cylinders to find missing dimensions of a cylinder given other dimensions. Students are given the image of a generic cylinder with marked dimensions for the radius, diameter, and height to help their reasoning about the different rows in the table..
While completing the table, students work with approximations and exact values of $\pi$ as well as statements that require reasoning about squared values. The final row of the table asks students to find missing dimensions given an expression representing volume that uses letters to represent the height and the radius. This requires students to manipulate expressions consisting only of variables representing dimensions.
Encourage students to make use of work done in some rows to help find missing information in other rows. Identify students who use this strategy and ask them to share during the discussion.
Give students 6–8 minutes of work time followed by a whole-class discussion.
If short on time, consider assigning students only some of the rows to complete.
Conceptual Processing: Eliminate Barriers.Allow students to use calculators to ensure inclusive participation in the activity.
Each row of the table has information about a particular cylinder. Complete the table with the missing dimensions.
|diameter (units)||radius (units)||area of the base (square units)||height (units)||volume (cubic units)|
|$b$||$\pi \boldcdot b\boldcdot a^2$|
Select previously identified students to share their strategies. Ask students:
- “What patterns did you see as you filled out the table?” (Sample reasoning: Rows that had the same base area were easier to compare because their volume was the base area times height.)
- “Look at rows 1 and 3 in the table. How did having one row filled out help you fill out the other more efficiently?” (If the base areas were the same, then the radius and diameter must be the same also.)
- “How did you reason about the last row?”
14.4: Find the Height
Cool-down: 5 minutes
This cylinder has a volume of $12\pi$ cubic inches and a diameter of $4$ inches. Find the cylinder's radius and height.
Student Lesson Summary
In an earlier lesson we learned that the volume, $V$, of a cylinder with radius $r$ and height $h$ is
$$V=\pi r^2 h$$
We say that the volume depends on the radius and height, and if we know the radius and height, we can find the volume. It is also true that if we know the volume and one dimension (either radius or height), we can find the other dimension.
For example, imagine a cylinder that has a volume of $500\pi$ cm3 and a radius of 5 cm, but the height is unknown. From the volume formula we know that
$$500\pi=\pi \boldcdot 25 \boldcdot h$$
must be true. Looking at the structure of the equation, we can see that $500 = 25h$. That means that the height has to be 20 cm, since $500\div 25 = 20$.
Now imagine another cylinder that also has a volume of $500\pi$ cm3 with an unknown radius and a height of 5 cm. Then we know that
$$500\pi=\pi\boldcdot r^2\boldcdot 5$$
must be true. Looking at the structure of this equation, we can see that $r^2 = 100$. So the radius must be 10 cm.