If the length of a cube’s edge is $a$, what’s the distance between opposite corners of the cube?
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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 $\sqrt{3}a$.