Session 7, Part B:
Area of a Circle

In This Part: Transforming a Circle | Examining the Formula

Let's further examine the formula for area of a circle, A = • r2. How do we interpret the symbols r2? If r is the radius of a circle, then r2 is a square with sides of length r. Examine the circles below. A portion of each circle is covered by a shaded square. We can call each of these squares a radius square.

Problem B5

Use the circles (PDF document) to work on this problem. For each circle, cut out several copies of the radius square from a separate sheet of centimeter grid paper. Determine the number of radius squares it takes to cover each circle. You may cut the radius squares into parts if you need to. Record your data in the table below.

Circle

Area of Circle

 1 2 3

Circle

Area of Circle

 1 6 36 36 • A little more than 3. 2 4 16 16 • A little more than 3. 3 3 9 9 • A little more than 3.

Problem B6

 a. What patterns do you observe in your data? b. If you were to estimate the area of any circle in radius squares, what would you report as the best estimate?

 Problem B7 Does the activity of determining the number of radius squares it takes to cover a circle provide any insights into the formula for the area of a circle?

 Video Segment Watch this segment to see how Janet and David use this method to make sense of the formula for the area of a circle. They cover a circle with four squares with sides equal to the radius of the circle. Then they cut up the squares to find out how many of the four squares fit into the circle. Watch this segment after you've completed Problems B5-B7. Did you come up with similar observations? If you are using a VCR, you can find this segment on the session video approximately 17 minutes and 13 seconds after the Annenberg Media logo.

 Problem B8 When you enlarge a circle so that the radius is twice as long (a scale factor of 2), what do you think happens to the circumference and the area? Do they double? Experiment by enlarging circles with different radii and analyzing the data.

 Problem B9 Experiment by enlarging a circle by a scale factor of 3, by a scale factor of 2/3, and by a scale factor of k. Generalize your findings. Problem B10 If a circle has a radius of 5 cm and the margin of error in measurement is 0.2 cm, what is a reasonable approximation for the area of the circle?

 Video Segment Circles have been widely used throughout the history of humankind in many different applications and human endeavors. In this segment, Afshan Bokhari explains the significance of circles in Islamic tradition, particularly in design and architecture. She shows us how the circle has been used as both an aesthetic element and a structural element in the building of mosques. Do you know of any other historical examples of the use of circles? If you are using a VCR, you can find this segment on the session video approximately 22 minutes and 27 seconds after the Annenberg Media logo.

 Problems B5-B10 adapted from Lappan, G.; Fitzgerald, W.M.; Phillips, E.D.; Fey, J.T.; and Friel, S.N. Connected Mathematics Program Covering and Surrounding. p. 140. © 1996 by Michigan State University. Published by Prentice Hall. Used with permission of Pearson Education, Inc.

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 Session 7: Index | Notes | Solutions | Video