A rectangular box has dimensions $3$ inches by $4$ inches by $5$ inches. What is the sum of the lengths of its four space diagonals?

1. Each space diagonal has length $\sqrt{3^2+4^2+5^2}=\sqrt{50}=5\sqrt2$.
2. A rectangular box has four equal space diagonals, so their total length is $4(5\sqrt2)=20\sqrt2$ inches.