I was at a dinner the other night at a restaurant when we moaned about the fact that the rectangular table was so long that those of us at one end could hardly speak to those at the other. Restaurant designers are acutely aware of the fact for a given perimeter, a circle occupies the greatest area of all possible shapes.
Talk therefore turned to circles and my colleague said the volume of a sphere is obtained by integrating the area of a circle.
I did a quick differentiation in my mind- 4/3 pi r^3 differentiates to 4 pi r^2, so it was easy to prove him wrong.
I just found out why- the area of a sphere is given by 4 pi r^2.
But hey, did you know this? For a cone, sphere and cylinder, where radius equals height:
Volume of a cone= 1/3 pi r^2. h= 1/3 pi r^2.2r= 2/3 pi.r^3 X 1
Volume of a sphere= 4/3 pi r^3= 2/3 pi. r^3 X 2
Volume of a cylinder= pi.r^2.h= pi.r^2.2r= 2/3 pi.r^3 X 3
Therefore the volume of equiradial cone, sphere and cylinder has a ratio of 1:2:3.
The relationshiip between the cone and cylinder is a bit obvious, but I had no idea that you could make a sphere out of 2 cones or a cylinder out of 1.5 spheres. Did you?
