I was trying to guess how many jellybeans were in a jar. If the jar was a glass cube or a glass box, I would just count width × length × height. With a cylinder however, I figure I could still use this method, but just minus the ratio of circle in square. So I became curious to see if the ratios were the same when reversed.
Take a circle, any radius. Now grow a square inside of it, until the four corners of the square touch the perimeters of the circle.
Will the area of the square always hold the same ratio to the area of the circle? If yes, what is that ratio?
Now vice versa
Take a square, any radius. Now grow a circle inside of it, until the circle perimeter touches each side of the square.
Will the area of the circle always hold the same ratio to the area of the square? If yes, what is that ratio?