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.
Lesson overview
 8.1 Warmup: Irrigating a Field (5 minutes)

8.2 Activity: Making a Polygon out of a Circle (20 minutes)
 There is a digital applet in this activity.

8.3 Activity: Making Another Polygon out of a Circle (10 minutes)
 There is a digital applet in this activity.

8.4 Activity: Tiling a Table (5 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 8.5 Cooldown: A Circumference of 44 (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s rearrange circles to calculate their areas.
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.
Required materials:
 scissors
 markers
 blank paper
 cylindrical household items
 glue or glue sticks
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.
Glossary :
 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 \(s \cdot s\), or \(s^2\).
 Acess the complete Grade 7 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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