How do you find the length width and the height of a rectangular prism if you already have your surface area?

Answer 1

You cannot find a unique solution because blocks of different shapes and volumes can still have the same surface area.

You can find the dimensions of a cube with the given surface area; it is the largest volume of a rectangular prism for a given surface area.

A cube has six equal square sides. Take the total surface area "At", divide this by 6 to get the area of one side "As":

At / 6 = As

The length Ls of one side of the cube is the square root of As .

Ls = sqrt (As ) = sqrt (At / 6 )

All the edges of a cube are of course the exact same length.

Another solution is a very thin square prism (almost a sheet but with a finite thickness) with the top and bottom just slightly less than half the surface area; the remainder has to be spread out along the edge which can be arbitrarily narrow.

Theoretically you can have a very long rectangular prism and so very large values for the length if the height and width are very small and still have the surface area set out in the problem.