In this activity, students estimate the area of various circles on a grid and see that the relationship between the diameter and area of a circle is not proportional. This echoes the earlier exploration comparing the length of a diagonal of a square to the area of the square, which was also not proportional.

Each group estimates the area of one smaller circle and one larger circle. After estimating the area of their circles, students graph the class’s data on a coordinate plane to notice that the data points curve upward instead of making a straight line through the origin. As students measure multiple circles and notice patterns in their measurements, they express regularity in repeated reasoning (MP8).

In the digital version of the activity, students use an applet to graph the relationship between diameter and area of a circle. The applet allows students to enter values into a table and see the points plotted on a coordinate plane. The digital version may help students graph quickly and accurately so they can focus more on the mathematical analysis.

The goal of this discussion is to help students see and express that the relationship between diameter and area of a circle is not proportional.

First, display the image of the coordinate plane for all to see. Ask students to share the diameter and estimated area of each circle. Plot these values on the grid.

Next, direct students' attention to the reference created using Collect and Display. Ask students to share what they notice about this graph of diameter and area and how it compares to the graph of diameter and circumference that they saw earlier. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.

The key takeaways are:

• The graph shows that the area of the circle goes up faster the bigger the diameter; it curves upward.
• The relationship between the diameter and area of circles is not a proportional relationship.

If desired, emphasize that the relationship is not proportional by having students add a column to their table of measurements and calculate the quotient of the area divided by the diameter. Here is a table of sample values.

Ask students:

• “Are the values for area divided by diameter close to the same value for each row?” (No)
• “What does this tell us about the relationship between diameter and area of a circle?” (There is no constant of proportionality. It is not a proportional relationship.)

In this activity, students cut and rearrange parts of a circle to approximate a parallelogram. Then they use what they know about finding the area of a parallelogram to develop a formula for the area of a circle.

Students see that the area of the parallelogram would be calculated by multiplying half of the circle’s circumference times its radius. Since students are not familiar with the process of writing proofs, it is necessary to walk them through writing the justification that uses the formula for the area of the parallelogram to develop the formula for the area of the circle. As students successively cut the circle pieces in half, rearrange them, and notice that the result more closely resembles a parallelogram, they look for regularity in repeated reasoning (MP8). As students use what they know about the area of a parallelogram to formulate the area of the circle, they make use of structure (MP7).

The purpose of this activity is for students to consider a different way to cut and reassemble a circle into something resembling a polygon in order to calculate its area. This time the polygon is a triangle, but the area of the circle can still be found by multiplying times the circumference times the radius.

In this activity, students write their own justification for the area of a circle. Monitor for students who have different explanations, including expressions such as:

As students create their own derivations for the area of a circle and compare them with others’, they construct arguments and critique reasoning (MP3).

A note about precision and approximation:

If the bands making up the circle really did not stretch, then they would not form rectangles when they are unwound because the circumference of the inner circle is not the same as the circumference of the outer circle in each band. A rectangle is an appropriate approximation for the shape in terms of calculating its area.

The purpose of this discussion is for students to collaboratively create an informal derivation for the area of a circle based on comparing it to the area of a triangle.

Ask students:

• “What polygon does the unrolled shape most resemble?” (a triangle)
• “How do we calculate the area of a triangle?” ( base times height)

Next, ask students to trade papers with a partner and discuss each other’s work for finding the area. Display these prompts for all to see:

• “I agree with this step because . . . .”
• “I disagree with this step because . . . .”
• “Can you explain how . . . ?”
• “Can you explain why . . . ?”
• “You should expand on . . . .”

If time permits, ask students to revise their explanations based on their partner’s feedback.

Display 1–2 approaches from previously selected students for all to see. Connect the responses to the learning goals by asking questions such as:

• “Where does the expression come from?”
• “What kinds of details or language helped you understand the display?”
• “Are there any details or language that you have questions about?”

The key takeaway is that the area of a circle can be found by multiplying .