 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 7, Part B:
Area of a Circle (60 minutes)

In This Part: Transforming a Circle | Examining the Formula

 In Session 6, we found the areas of different polygons (parallelograms, triangles) by dissecting the polygons and rearranging the pieces into a recognizable simpler shape. In this case, we transformed a parallelogram into a rectangle by slicing a triangle off one end and sliding it along to fit into the other end. In doing so, we established that the area of the parallelogram was the same as the area of the equivalent rectangle (its base multiplied by the perpendicular height). Can we use the same technique and transform a circle into a rectangular shape? Use a compass and draw a large circle. Fold the circle in half horizontally and vertically. Cut the circle into four wedges on the fold lines. Then fold each wedge into quarters. Cut each wedge on the fold lines. You will have 16 wedges. Tape the wedges to a piece of paper to form the following figure: Notice that we have a crude parallelogram with a height equal to the radius of the original circle and a base roughly equal to half the circumference of the original circle.  Problem B1 How does the area of the figure compare with the area of the circle? Problem B2 The scalloped base of the figure is formed by arcs of the circle. Write an expression relating the length of the base b to the circumference C of the circle.  Is b equal to the circumference of the circle? You may need to re-form the wedges back into a circle and then back again into a parallelogram.    Close Tip Is b equal to the circumference of the circle? You may need to re-form the wedges back into a circle and then back again into a parallelogram. Problem B3 Write an expression for the length of the base b in terms of the radius r of the circle.  C = d or C = 2 r. Substitute one of these expressions for C in your equation in Problem B2.    Close Tip C = d or C = 2 r. Substitute one of these expressions for C in your equation in Problem B2. Problem B4 If you increase the number of wedges, the figure you create becomes an increasingly improved approximation of a parallelogram with base b and height r. Write an expression for the area of the rectangle in terms of r. Think about how the activity involving wedges helps explain the area formula of a circle, A = • r2.

 Problems B1-B4 adapted from Bass, L.; Hall, B.; Johnson, A.; and Wood, D. Geometry: Tools for a Changing World. © 1998 by Prentice Hall. Used with permission. All rights reserved.   Session 7: Index | Notes | Solutions | Video