Express the volume of a cube as a function of its surface area, S?

Answer 1

The volume of a cube as a function of the length of a side is
V = x^3
The Surface area of a cube as a function of the length of a side is
SA = 6x^2
Solving for the length of one side in the SA equation gives us
(SA/6)^1/2 = x
Now substitute that into the volme equation for x
V = (SA/6)^(3/2)
Or one sixth of the surface area to the 3/2ths power.
Or one sixth of the surface area cubed and square rooted.
Or the square root of one sixth of the surface area cubed.

Answer 2

Note that the volume of a cube is

V = x^3 (where x is the length of a side)

and the surface area of a cube is

S = 6x^2 (x^2 is each face of the cube, and there are 6 faces)

So all we have to do is algraically change V.

V = x^3 = x * x^2
= x * 6x^2/6
= x * S/6

But, since S = 6x^2, then S/6 = x^2 and x = sqrt(S/6)

So

V = sqrt(S/6) * S/6

Answer 3

S = 6 times the area of a side = 6e^2, where is the length of an edge.

e = sqrt(S/6)

Then the volume is e cubed, so

V(S) = e^3 = (sqrt(S/6))^3 = (S/6)^(3/2)