# Lesson 15

Area of a Circle

### Lesson Narrative

This lesson develops the formula for the area of a circle. Students begin estimating areas of different-sized circles to see that area and diameter do not have a proportional relationship. They then discover that unrolling a circle into a triangle shape suggests that area and circumference are related by the equation $$A = \frac{1}{2} \boldcdot C \boldcdot r$$ where $$A$$ is the area, $$C$$ is the circumference, and $$r$$ is the radius. Optionally, students can cut a circle into sectors and rearrange the sectors into a shape resembling a parallelogram to find a similar formula. Students look for and express regularity in repeated reasoning (MP8) when they informally derive the formula after estimating areas of circles.

When we say “area of a circle” we technically mean “area of the region enclosed by a circle.” However, “area of a circle” is the phrase most commonly used.

This lesson presents a few options for informally deriving a formula for the area of a circle.

• One sequence allows students to spend additional time estimating areas of circles and noticing that area is not proportional to diameter using the activity Estimating Areas of Circles. Then, students informally derive a formula for the area of a circle by unrolling a circle into a triangular shape using the activity Making Another Polygon out of a Circle.
• Another sequence involves shortening the time spent on the activity Estimating Areas of Circles by using the digital version of the activity and giving each group fewer examples to work with. Then, students create a visual display informally deriving a formula for the area of a circle by dissecting a circle into sectors and rearranging the sectors into a shape resembling a parallelogram.

### Learning Goals

Teacher Facing

• Create a table and a graph that represent the relationship between the diameter and area of circles of various sizes, and justify (using words and other representations) that this relationship is not proportional.
• 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 investigate the areas of circles.

### Required Preparation

For the first activity, you will need the Estimating Areas of Circles blackline master. Prepare 1 copy for every 12 students. (Each group of 2 students gets one of the pages.)

If you are using the optional activity Making a Polygon out of a Circle, 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 and other materials to make a visual display.

### Student Facing

• I know the formula for area of a circle.
• I know whether or not the relationship between the diameter and area of a circle is proportional and can explain how I know.