Slices of Pi

This picture demonstrates the famous formula:

Area of a circle = Pi  r²

^{Here's how to see it:}

^{Look at the slices in the circle - each slice is "r," the radius, long on a side.}

^{Here is what we know before we start:}

• The circumference of the circle is Pi times the diameter.
• And the diameter is 2 times the radius.
• So the circumference of the circle is 2 r Pi 

Here is what we do:

• So now we take the slices and rearrange them into a rectangular like thing.
• The height is "r" because it is the same length as the radius.

The tricky bit:

• Think about the width of our rectangle-ish shape.
• The bottom edge is the "pie crust" of half of the circle.
• Half the crust on the top edge, half on the bottom.
• So the width must be HALF the circumference, or Pi  times the radius (or Pi times half of the diameter).
• Take a minute to think about it - it's a lttle tricky.  
• What would be a good estimate to the area of this rectangle-ish shape be?  
• Height times width or     r x (Pi x r)   or...

Area = Pi  r²

^{To see this more exactly, try to imagine infinite slices (rather than 16) getting infinitely thin.}

[video:https://youtu.be/Gat6PTId43Y]

Open a PDF version that you can cut out yourself