Surface areas of two cubes

Question

There are two cubes. The points $P$ and $Q$ are both on the small cube: $P$ is a point in the centre of one of the faces, and $Q$ is a corner on the opposite face.

The second cube has sides of length $|PQ|$. What is the surface area of the large cube divided by the surface area of the smaller cube?

I drew both of them and know that the SA of the large cube is $6{QP}$. My problem is with the small cube. In the small cube, if I would connect points $P$ and $Q$, they would for a right angled triangle and line $PQ$ is the hypotenuse. However what is the purpose of this line, how can we find the SA using this?

Answer 1

The area' ratio $R$ is the square of the length's ratio $r$. Consider that the cube has sidelength 1.

Let $P'$ be the center of the opposite face of $P$. It is clearly the orthogonal projection of $P$ onto this face. Thus, triangle $PP'Q$ is rectangle in $P'$ with $PP'=1$ and $P'Q=\sqrt{2}/2$ (remember that the diagonal of a square with side 1 is $\sqrt{2}$).

Pythagoras' theorem, applied to triangle $PP'Q$ gives:

$PQ^2=1^2+(\sqrt{2}/2)^2=3/2$. Thus $PQ=\sqrt{3/2}$, giving $r=PQ/1=\sqrt{3/2}$.

Therefore, the ratio of lengths is $r=\sqrt{3/2}/1=\sqrt{3/2}$

Thus the ratio of areas is the square of the previous ratio : $R=r^2=3/2.$

Answer 2

Take the cube $[0,1]^3$ as small cube for example (without loss of generality as you may pick your coordinate system any way you want for this task). It has surface area $6$ as its six sides have area $1$ each. Pick e.g. $Q=(0,0,0)$ and $P=(1,\frac12,\frac12)$. Then the distance is $$\lvert P-Q\rvert=\sqrt{1+\frac14+\frac14}=\sqrt{\frac32}$$ So the surface area of the large cube is $6\cdot\frac32=9$ and the quotient between these suface areas is $$\frac96=\frac32$$