Question

Express the edge length of a cube as a function of the cube's diagonal length d. Then express the surface area & volume of the cube as a function of the diagonal length.

Answer 1

Each side of the cube has length x

On one face of the cube, the diagonal is found using Pythagorean's Theorem a^2 + b^2 = c^2

Since this is a cube, a= b = x. And c^2 = sqrt(x^2 + x^2) , so c= sqrt(2x^2)

Now d, the diagonal of the cube, can be found side it is formed by a triangle with a side of length x and a side of length sqrt(2x^2)

So d^2=( sqrt(2x^2))^2 + x^2
d = sqrt(3x^2)

Solving for x gives x=sqrt((d^2)/3)

Each face has an area of x^2 = (d^2)/3

Surface area = 6 x ((d^2)/3). ( there are 6 faces on a cube)

And volume is x^3 = (sqrt((d^2)/3))^3

That was not easy to type up on an iPad!

Answer 2

wrong answer