mathematics - Baggage Problem: $1.5$-meter-long sword onto a train

This is the problem I came across reading the book The Art and Craft of Problem Solving. When I read this question I wasn't able to figure out the solution and I saw the solution after a while, but still I couldn't understand the solution as well.

Question:

Pat wants to take a $1.5$-meter-long sword onto a train, but the conductor won't allow it as carry-on luggage. And the baggage person won't take any item which greatest dimension exceeds $1$ meter. What should Pat do?

Solution:

This is unsolvable if we limit ourselves to two-dimensional space. Once liberated from Flatland, we get a nice solution : The sword fits into a $1 \times 1 \times 1$ -meter-box, with a long diagonal of $\sqrt{(1^2 + 1^2 + 1^2)}$ = $\sqrt{3} > 1.69$ meters.

Can anyone give me a clear explanation of this solution?

Answer

The sword fits in that 3D box, because the long diagonal is $1.69$ meter. The longest dimension of this box is 1 meter (length, width and height are all exactly 1 meter). The sword goes like the green line in the picture:

If Pat would use a flat case, it would be denied. In the most naive way, the box would be $1.5$ meter long and have a width of the sword itself. A more clever way to think is in the maximum dimensions, which are $1 \times 1$ meter. In that case the diagonal is $\sqrt{2} = 1.41$ meter long, which is not long enough. So the only option is to include the height into account.