How do you find the length of a cubes side if i only have the surface volume ration?

Answer 1

Lets assume the length of the cube edge is x.
Lets assume the surface area to volume ratio is y.

Now, the surface area of a single face of the cube (being a square) is x^2.
The total surface area of the cube is 6x^2 (because there are six faces).

Now, the volume of the cube is x^3.

The ratio of surface area to volume (y) can be written as:
y = 6x^2 / x^3

We need to reduce the equation and re-arrange it so that the equation gives the value of x rather than y. Lets simplify the right hand side by removing two "x"s from the top and the bottom. That gives:
y = 6 / x

Now we multiply both the left hand side and the right hand side by "x" to get:
xy = 6

Finally we divide both the left hand side and the right hand side by "y" to get:
x = 6/y

Just to be sure we can test the equation with a couple of examples:
1. A cube with length 1 has a volume of 1 and an area of 6. So the ratio would be 1/6. Putting this into the equation as y gets us back the value of x.

2. A cube with length 2 has a volume of 8 and an area of 24. So the ratio would be 3. Putting 3 into the equation for the value of y we again get the right value of x, 2.