As is well known to most students, the formula for the area of a circle with radius R is: πR². . But why? In this worksheet, we cut a given circle into rings and stack them up in an effort to explain the meaning behind the neat formula. Of course, there are other ways to derive the formula. I tried to make a physical model with ropes, but it proved very difficult. Hence, the GeoGebra simulation.
In this worksheet, we create two triangles: an isoscelese one and a right triangle. Please use all the information about the triangles to figure out the area of the circle. The base of the triangles is 2πR, as you may see. The height of the triangles is the radius of the given circle, isn't it?
To add more rings, please right click the slider RR and change the increment to a smaller number. You could also drag A, C, H, I, J to adjust the orientation and position of the triangles. To find out how it is created, please click "View:Contruction Protocol". Have fun! And do play with your math ideas.

Questions
1. How do you know what we get are indeed triangles?
2. Why are the no-base sides of the triangles not curves?
3. Can we say that all circles are simillar as we know them in everyday situations?
4. What is the consequence of saying all circles are similar?
Lingguo Bu (lgbu@siu.edu), Created with GeoGebra, August 1, 2011