Answer
$1.\ \ \mathrm{Length\:of\:side\:as\:a\:function\:of\:diagonal:}\ \ a(d)=\frac{d\sqrt{3}}{3}$
$2.\ \ \mathrm{Volume\:as\:a\:function\:of\:diagonal:}\ \ A(d)=2d^2$
$3.\ \ \mathrm{Surface\:area\:as\:a\:function\:of\:diagonal:}\ \ V(d)=\frac{d^3}{3\sqrt{3}}$
Work Step by Step
$1.$ To find the length of the space diagonal, we need to find the length of the face diagonal of the cube. Consider the length of side $\ a$. By applying the pythagorean theorem to the triangle $\ \mathrm{HCD}$, we have:
$(d_f)^2=a^2+a^2$
$\Rightarrow\ (d_f)^2=2a^2$
Now apply the pythagorean theorem to the triangle $\ \mathrm{FCH}\ $, to get the length of the space diagonal as:
$d^2=(d_f)^2+a^2$
$\Rightarrow\ d^2=2a^2+a^2$
$\Rightarrow\ a^2=\frac{d^2}{3}$
$\Rightarrow\ a=\frac{d}{\sqrt{3}}$
$\Rightarrow\ a=\frac{d\sqrt{3}}{3}\ \ $ by rationalizing the denominator.
So, the Length of side as a function of diagonal is, $\ \ a(d)=\frac{d\sqrt{3}}{3}$
$2.$ Formula for the surface area of a cube is $\ A=6a^2$.
So, the surface area as a function of diagonal is, $\ \ A(d)=6(\frac{d^2}{3})=2d^2$
$3.$ Formula for the volume of a cube is $\ V=a^3$.
So, the volume as a function of diagonal is, $\ \ V(d)=(\frac{d\sqrt{3}}{3})^3=\frac{d^3}{3\sqrt{3}}$
