The cube: Surface area and volume

Example

Calculate the diagonal $D$ of the cube of the figure if $a=10$.

If we apply the Pythagoras theorem in a square of side $10$, we can see that: $$\begin{gathered} d^2=2\cdot a^2=200 \\ d=10\sqrt{2} \end{gathered}$$

Now a rectanglular triangle with legs $a$ and $d$, and hypotenuse $D$ is formed . If we apply Pythagoras again: $$\begin{gathered} D^2=d^2+a^2=(10\sqrt{2}{)}^2+100 \\ D=\sqrt{3}\cdot =17,32 \end{gathered}$$

Then, $$\begin{gathered} D=\sqrt{3}\cdot a \\ A=6\cdot a^2 \\ V=a^3 \end{gathered}$$