How could you find a longest rod placed in a box if the dimensions are 2×4×5?

To do this, the longest rod would be the diagonal of the box.  Now, this is different than the diagonal of one side of the box.  I am talking about the diagonal that would go through the middle of the box.  I attached an example.

To, figure this out, you would need to find the "projected diagonal" of one side.  Like in the example, how long would the diagonal be if projected to the bottom of the box, the green line?  Here, we would make a right triangle with the bottom of the box, sides of 5 and 4.  So, we could calculate the length of the green line using Pythagorean's Theorem, so the length of the green segment, g, is ` ` `g = sqrt(4^2 + 5^2) = sqrt(41)`

Then, you would do the Pythagorean Theorem again using that diagonal, the diagonal through the middle of the box, g,  and the back right side of the box.  We can find the length of the red diagonal, r, using Pythagorean's Theorem again using the green segment, g, and the side of length 2. So, `r = sqrt(g^2 + 2^2) = 3sqrt(5)`

Or, the easy way would be to use the Pythagorean Theorem for 3D, which is  `x^2 + y^2 + z^2 = d^2`  

We know x =2, y = 4, and z = 5, 2, 4, and 5.  So, plugging in those, `d = sqrt(45) = 3sqrt(5)`