Relating Area to Circumference

5 min

Narrative

In this activity, students estimate the area of a circle by comparing it to a surrounding square. Students should recognize that the area of the circle is less than 640,000 m2, which is the area of the surrounding square.

Students who completed the optional activity “Covering a Circle” may recall that the area of a circle with radius rr is a little more than 3r23r^2. They can use this relationship to determine that the circle’s area is slightly greater than 480,000 m2.

Launch

Explain that some farms have circular fields because they use center-pivot irrigation. If desired, display these images to familiarize students with the context.

<p>Images of fields to find the area using circles.</p>

<p>Image of irrigating a field of cotton.</p>

Ask students to estimate the circular growing area (green region) in the image in their books or devices. Give students 1–2 minutes of quiet think time followed by whole-group discussion. 

Student Task

A circular field is set into a square with an 800-m side length.

A square where the vertical distance is labeled 800 meters.
A square where the vertical distance is labeled 800 meters. The largest possible circle is drawn inside the square with a line segment is that begins from the center of the circle to a point on the edge of the circle.
​​​

What is the field’s area? Record an estimate that is:

too low about right too high

Sample Response

Sample responses:

  • Too low: 320,000 m2 to 400,000 m2
  • About right: 480,000 m2 to 520,000 m2
  • Too high: 640,000 m2 or more

Sample reasoning:

  • The square around the field has an area of 800⋅800800 \boldcdot 800, or 640,000 m2. The circular field is a little less than that, around 80% or so.
  • The radius of the field is 400 m. A square with side lengths of 400 m has an area of 160,000 m2. It takes a little more than 3 of these squares to cover the circle, and 3⋅160,000=480,0003 \boldcdot 160,000 = 480,000.

Synthesis

Invite students to share their estimation strategies. To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”
Anticipated Misconceptions

Students might think the answer should be 640,000 m2 because that is the area of the square, not realizing that they are being asked to find the area of a circle. Ask them what shape is the region where the plants are growing.

Some students might incorrectly calculate the area of the square to be 6,400 m2 and therefore estimate that the circle would be about 5,000 m2.

Some students might try to use what they learned in the previous lessons about the relationship between the area of a circle and the area of a square with side length equal to the circle's radius. Point out that the question is asking for an estimate and answer choices all differ by a factor of 10.

Standards
Addressing
  • 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
  • 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

20 min

10 min

10 min