How Do We Know:

Fun Stuff:

- The Earth Revolves
- Atoms Exist
- Fossils are Old
- Distance to Stars
- The Earth Rotates
- The Earth is Round
- The Distance to the Sun
- The Speed of Light
- What Stars Are Made Of
- The Universe is Expanding

Fun Stuff:

- The Mandelbrot Set: An Introduction
- The Area of a Circle

Everyone knows that the area of a circle is π times the radius squared. I saw this demonstration of that fact when I was in Junior High School. It impressed me greatly with its simplicity and beauty.

I hope you enjoy it.

Start with a circle cut into quarters.

Now rearrange the pieces like this:

Next, cut one of the blue sections in half vertically, move the outer half to the other side. Center the blue parts above the red ones.

Here a picture really is worth a thousand words!

Now move the pieces together.

This object has the same area as the original circle. But finding it's area
isn't any simpler than what we started with. But we do know a few things:

The edges on the right and left are vertical and their length is the radius of
the circle, r.

The length of the bumpy path on the top, and the bottom, is half of the circumference of the
circle. The definition of π is:

π=C/D

D=2r (Diameter is twice the radius)

Combining these we get:

C=2πr.

This tells us that the length of each funny bumpy path is π*r

Now, imagine that you start with a circle cut into eighths instead of
quarters. Doing the same sort of rearrangements we get:

If we make 16 pie slices we get:

32 pie slices gives us:

64 results in:

128 looks like:

It is now apparent that as we increase the number of slices the resulting figure will get closer and closer to a perfect rectangle. The edges and the top and bottom have the same length they did before. The area is now easy to compute. It's just length times width.

Width = π * r

height = r

Area is π * r * r or π*r^{2}