Geometry of Circles

Extensions:

• To develop understanding of the area of a circle, have pairs of students cut up a paper plate using lines of symmetry through the center, just as one slices a pizza. Rearrange the slices as shown below. Students will realize that this configuration almost looks like a rectangle! How would this "rectangle" help in finding the area of a circle? [The width of the rectangle is equal to the radius of the original circle. The length of the rectangle is half of the circumference, since the entire circumference is both on the top and bottom. Therefore, the area is equal to the radius times half the circumference, or A = ½Cr. Because C = =πd and d = 2r, this formula becomes the more familiar A = πr2.]
• Allow students to use Geometer’s Sketchpad or other geometry software to create the constructions described in this lesson.
• Research how hat sizes were determined! Or, check out the web site of a company that makes and sells hats, and you might find a table like the one below. What is the relationship between men’s head measurement (in inches) and American hat sizes? Have students measure the circumference of their head, and divide it by π — the result is their hat size.
•  If students were to plot these points in a scatterplot, one reasonable line of best fit is y = 3.14x, indicating that the y‑value (head circumference) is approximately π times the x‑value (hat size).