Lesson 8

Relating Area to Circumference

Lesson Narrative

In the previous lesson, students found that it takes a little more than 3 squares with side lengths equal to the circle’s radius to completely cover a circle. Students may have predicted that the area of a circle can be found by multiplying \(\pi r^2\). In this lesson students derive that relationship through informal dissection arguments. In the main activity they cut and rearrange a circle into a shape that approximates a parallelogram (MP 3). In an optional activity, they consider a different way to cut an rearrange a circle into a shape that approximates a triangle. In both arguments, one side of the polygon comes from the circumference of the circle, leading to the presence of \(\pi\) in the formula for the area of a circle.


Learning Goals

Teacher Facing

  • Generalize a process for finding the area of a circle, and justify (orally) why this can be abstracted as $\pi r^2$.
  • Show how a circle can be decomposed and rearranged to approximate a polygon, and justify (orally and in writing) that the area of this polygon is equal to half of the circle’s circumference multiplied by its radius.

Student Facing

Let’s rearrange circles to calculate their areas.

Required Preparation

You will need one cylindrical household item (like a can of soup) for each group of 2 students. The activity works best if the diameter of the item is between 3 and 5 inches.

If possible, it would be best to give each group 2 different colors of blank paper.

Learning Targets

Student Facing

  • I can explain how the area of a circle and its circumference are related to each other.
  • I know the formula for area of a circle.

CCSS Standards

Addressing

Glossary Entries

  • area of a circle

    If the radius of a circle is \(r\) units, then the area of the circle is \(\pi r^2\) square units.

    For example, a circle has radius 3 inches. Its area is \(\pi 3^2\) square inches, or \(9\pi\) square inches, which is approximately 28.3 square inches.

  • squared

    We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).

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