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