If the length of a cube’s edge is $a$, what’s the distance between opposite corners of the cube? Show hint

If you’re not sure what to do, ask yourself, what can I work out from the given information? Or work backwards from the goal: ask yourself, what information would I need to work out first in order to find the diagonal?

Let the distance between opposite corners be $c$. First we need to consider the length of the diagonal of a face of the cube. Let’s call this length $b$. $b$ is the hypotenuse of a right-angled triangle with both other sides of length $a$: Hence, $b^2=a^2+a^2=2a^2$. $c$ is the hypotenuse of another right-angled triangle with the other two sides of lengths $a$ and $b$: Therefore \begin{align*} c^2&=a^2+b^2\\ c&=\sqrt{a^2+2a^2}\\ &=\sqrt{3}a \end{align*} So the distance is $\sqrt{3}a$.