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 .

In this activity, students calculate the area of circles given their radius or diameter. For the problem where the radius is given, students practice applying the formula . For the problem where the diameter is given, students must think about what extra steps are required. This is similar to the thinking they did previously when they were given a circle’s radius and asked to calculate its circumference.

Because of the properties of multiplication, there is some flexibility when it comes to strategies for calculating circumference given the radius, such as:

• Multiplying the radius by 2 to find the diameter, then multiplying the diameter by .
• Multiplying the radius by to directly calculate the circumference.
• Multiplying the radius by first and then multiplying that product by 2.

However, because the formula for area of a circle involves squaring the radius, there is less flexibility possible for calculating the area given the diameter. Students are most likely to be successful using the strategy:

• Dividing the diameter by 2 to find the radius, then using the formula .

During the whole-class discussion, students critique a response that is intentionally incorrect and improve it by clarifying meaning, correcting errors, and adding details (MP3). 

The purpose of this discussion is to highlight students’ strategies for dealing with being given the diameter of the beaded medallion, rather than its radius.

First, invite students to share their reasoning about the area of the hoop drum. Once the class is in agreement on the answer, move the discussion on by asking “What was different about the second problem compared to the first problem?” While there are several differences students may notice, the important discussion point is that the diameter was given, rather than the radius.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response for finding the area of the medallion by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  “To find the area of the medallion, I multiplied  times 6². Then I realized that 6 was the diameter and not the radius, so I divided my answer by 2.”
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

The key takeaway is that it is important to divide the diameter by 2 first to find the radius before applying the formula .