Perfect Cuboid Problem

Remember the pythagorean theorem, A2 + B2 = C2? The three letters correspond to the three sides of a right triangle. In a Pythagorean triangle, and all three sides are whole numbers. Let's extend this idea to three dimensions. In three dimensions, there are four numbers. In the image above, they are A, B, C, and G. The first three are the dimensions of a box, and G is the diagonal running from one of the top corners to the opposite bottom corner.

knownhappyman68
Aug 29, 2017

#1**+1 **

We can break this down in parts

The diagonal distance [in the same plane] across the bottom of the box will be

f =sqrt [ (d^2 -a^2) + (e^2 - a^2) ] = sqrt [ b^2 + c^2]

And f^2 = (b^2 + c^2)

Then....the height of the box = a = sqrt [ e^2 - c^2 ]

And a^2 = (e^2 - c^2)

So....by the Pythagorean Theorem....g =

sqrt [ f^2 + a^2] =

sqrt [ (b^2 + c^2) + (e^2 - c^2) ] =

sqrt [ b^2 + e^2]

CPhill
Aug 29, 2017