The Volume of a Cylinder

10 min

Teacher Prep
Setup
Refamiliarize students with circle vocabulary. 3 minutes of quiet work time, follow with a whole-class discussion.

Narrative

The purpose of this Warm-up is for students to review how to compute the area of a circle, an idea developed in grade 7. This Warm-up also gives students an opportunity to revisit language and calculations related to circles in preparation for finding the volume of a cylinder later in the lesson.

Students begin the activity identifying important features of a circle, including its radius and diameter. They use this information and the formula for the area of the circle to choose expressions from a list that are equivalent to the area of the circle. In the final question, students are given the area of the circle and are asked to find the corresponding radius.

Launch

Display the diagram from the Task Statement for all to see, and ask students:

  • “Name a segment that is a radius of circle AA.” (ACAC, ADAD, and ABAB, or these segments with the letters reversed, are all radii.) Review the meaning of the radius of a circle.
  • “What do we call a segment like BCBC, one with endpoints on the circle that contains the center of the circle?” (diameter) Review the meaning of the diameter of a circle.
  • “What is the length of segment ABAB?” (4 units) Review the fact that all radii of a circle have the same length.

Give students 3 minutes of quiet work time, and follow with a whole-class discussion. As students are working, select students who can explain why 16π16\pi , π42\pi4^2, and “approximately 50” square units represent the area of the circle.

Student Task

Here is a circle. Points AA, BB, CC, and DD and segments ADAD and BCBC are drawn.

A circle with center A. Points C, D, and B are on the circle. Segment C B contains A. Segment D A, has length 4.

  1. What is the area of the circle, in square units? Select all that apply.
    1. 4π4\pi
    2. π8\pi 8
    3. 16π16\pi
    4. π42\pi 4^2
    5. approximately 25
    6. approximately 50
  2. If the area of a circle is 49π49\pi square units, what is its radius? Explain your reasoning.

Sample Response

  1. c, d, and f. Since the radius is 4, the area of the circle is π⋅42=16π\pi\boldcdot 4^2=16\pi. This is approximately 50.3 square units.
  2. 7 units. Sample reasoning: The formula for area is times the square of the radius, and the area of this circle is 49 square units. So the square of the radius is 49π49\pi, and the radius is 7 units because 49=7249=7^2.
Activity Synthesis (Teacher Notes)

The purpose of this discussion is to make sure students remember that the area of a circle can be found by squaring its radius and multiplying by π\pi.

Ask previously selected students to share answers to the first question and explain why each of the solutions represents the area of the circle. If not brought up during the discussion, tell students that sometimes it is better to express an area measurement in terms of π\pi. Other times it may be better to use an approximation of π\pi, like 3.14, to represent the area measurement in decimal form. In this unit, we will often express our answers in terms of π\pi.

Standards
Building On
  • 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
  • 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

15 min

10 min