The cube: Surface area and volume

The cube is a figure composed by $$6$$ square faces, as we can see in the following picture


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: $$$d^2=2 \cdot a^2=200\\ d= 10 \sqrt{2}$$$

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

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