How do you find the surface area to volume ratio of a sphere?

Question

Also, what would the dimensions of my sphere have to be in order to get the lowest possible surface area to volume ratio. (Meaning the volume is high but surface area is low :-) )

Answer 1

Let's set up the ratio using the formulas for SA and V of a sphere, then simplify it:

 

SA = 4(pi)r²      V = (4/3)(pi)r³

 

  SA           4(pi)r²                  (3/4)4(pi)r²          3

 -----   =   -------------     =        ---------------    =   ----   

   V           (4/3)(pi)r³                   (pi)r³               r

 

Since the radius, r, is the denominator, the larger the radius is, the smaller the SA/V ratio will be.  For example, the ratio for a sphere with a radius of 1 is 3/1.  The ratio for a sphere with a radius of 10 is 3/10.  So the sphere with the lowest SA/V ratio would have a radius of infinity.

 

Does this all make sense?