# 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.

### 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.

### 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 $$s^2$$.