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

d = sqrt(3s^2) where s is the length of the side. Solving for s,

3s^2 = d^2 iff
s^2 = d^2 / 3 iff
s = sqrt(d^2 / 3)
= d / sqrt(3) or d sqrt(3) / 3

Surface area of the cube = 6 s^2. Thus,
A = 6 (d / sqrt(3))^2
= 6d^2 / 3
= 2d^2

Volume = s^3. Thus,
V = (d / sqrt(3))^3
= d^3 / 3sqrt(3)
= d^3 sqrt(3) / 9

Answer 2

d = sqrt(3s^2) ,

3s^2 = d^2
s^2 = d^2 / 3
s = sqrt(d^2 / 3)
= d / sqrt(3)
now we will find surface area
therefore
Surface area of the cube = 6 s^2.
Area = 6 (d / sqrt(3))^2
= 6d^2 / 3
= 2d^2
now we shall find volume
V = s^3.
V = (d / sqrt(3))^3
= d^3 / 3sqrt(3)
= d^3 sqrt(3) /9
hope this helps