Unit 3, Lesson 8
Relating Area to Circumference
Grade 7
Preparation
Lesson Narrative
In this lesson, students see that the area of a circle can be found by multiplying . They look at informal dissection arguments to derive that relationship.
First, students cut and rearrange a circle into a shape that approximates a parallelogram (MP8). The next activity is optional because it shows a different way to cut and rearrange a circle into a shape that approximates a triangle. In each case, the area of the polygon is equal to . Using algebraic reasoning, students construct and critique arguments that this is equivalent to or (MP3). The little “2” up in the air is pronounced squared and means that the value of r is multiplied by itself. Finally, students apply the formula to solve problems.
- 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.
Let’s rearrange circles to calculate their areas.
Required Materials
Required Preparation
Consider copying the blackline master onto different colors of paper. If different colors of copy paper are not available, the blackline master can be copied onto white paper and then construction paper can be used for the other sheet that the pieces are glued onto.