brent@itm.UUCP (08/16/83)
As Lew pointed out, a sphere with radius r and a cylinder with radius
r and height 2r have the same surface area to volume ratio: 3/r
Now try a cube of edge length 2r. 6*(2*r)^2 / (2*r)^3 = 3/r
The same as the cube and the cylinder. What's wrong?
The ratio of surface area to volume depends on what you call r.
Example: try a unit cube with edge length r. The S/A ratio is
6/r. Compare this with the 3/r answer obtained above.
The proper phrasing of the constraint is something like "The sphere
gives the minimum surface area of any shape *for any given volume*"
Compute the surface areas for a cube, a cylinder and a sphere
of volume 1. It comes out something like:
Shape Volume Area
Cube 1 6
Cylinder 1 5.54
Sphere 1 4.84
Indeed the sphere is the minimum area for the given volume enclosed.
The previous results of 3/r resulted from clever choices for r,
thus comparing the ratios of shapes enclosing different volumes.
Brent Laminack (msdc!itm!brent)