2006-09-13 03:31:27 · 5 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
The first and third answers on here are pretty good. I just visited about 10 sites looking for a simple explanation and the best one I found I am putting in the link below.
BTW, the constant you are referring to is pi which is approximately 3.14159... The constant that you mentioned phi is entirely different and is 1.618... and is often called the golden ratio. Entirely differnent.
2006-09-13 04:01:05 · answer #1 · answered by Will 4 · 1⤊ 0⤋
rt66lt is right. It may be easier first to compute the volume. The volume of a slice with z-coordinate z and a very thin thckness dz is pi*(1-z^2). Now integrate this from -1 to 1 and you get 4/3 pi r^3.
The next step is to imagine the phere as build from thin cones. The surface of the sphere is the sum of the ground areas of all the cones. Since the volume of a cone with height r and ground area A is 1/3 r A, the surface of the sphere is (4/3 pi r^3 ) / (1/3 r) which is 4 pi r^2.
2006-09-13 10:55:19 · answer #2 · answered by helene_thygesen 4 · 2⤊ 0⤋
Because a sphere is a circle that is rotated through three-dimensional space. If you take calculus, there is a method where you can mathematically rotate a circle through the third dimension and you wind up with the formula for the surface area of a sphere. I forget exactly how this is done, it's been about twenty years since I had calculus, but I remember doing it.
2006-09-13 10:39:27 · answer #3 · answered by The Doctor 7 · 0⤊ 0⤋
Here's a quick demonstartion of why the area of a sphere is 4x pi x r^2
Its not a complete proof, but it does demonstrate that the formula is in the right neck of the woods. A proof would go into quite high level calculus.
2006-09-13 10:58:52 · answer #4 · answered by robcraine 4 · 2⤊ 0⤋
I think you are thinking of two different values for r. r must be the same.
2006-09-13 10:42:20 · answer #5 · answered by banjuja58 4 · 0⤊ 1⤋