Area of a circle from what we already know
An interesting surprise
In this section we will explore how to determine the area of a circle by dividing it into sections and then rearranging these sections to form a shape with an easily recognizable area.
Follow the instructions and have fun!
1. Slider 1
The applet below shows a circle divided into 4 sections.
As we move slider 1 the number of sections increases by multiples of 4.
Move slider 1 to 3 and notice that there are now 12 sections in the circle.
We will explore what happens when we play with the other two sliders in the sections below.
2. Slider 2
Set slider 1 to 5 (The circle will have 20 sections).
Move slider 2 all the way to the left and describe what happens.
You can change the value of slider 1 to greater numbers to help in your description.
3. Slider 3
Now move slider 3 all the way down slowly and describe what happens once it gets to the bottom. Give as much detail as possible.
4. What shape do you see?
As slider 3 moved down with a large number of sections for slider 1, what shape is created by rearranging the sections?
5. Now, what is the area?
If you answered parallelogram in the previous question, you are correct!
Write the formula for the area of a parallelogram.
6. Connecting to the circle part 1
The area of a parallelogram is
The height of the parallelogram is the same as...
7. Connecting to the circle part 2 - the base
The area of a parallelogram is
The applet shows the base =
Which of the following best describes why this is true?
8. A little algebra
Simplify the right side of this equation. This is the area of the parallelogram:
Comment on your simplified equation.