Is there a rectangular pyramid whose altitude, slant height, and ALL edge lengths are all integers, or rationals?

Question

f so, have any suggestions on methods to find all such pyramids?

but besides guess-and-check don't know where to begin.

Background: I am a high school math teacher and got curious about this when generating problems. Not a homework set for me.

Answer 1

It is an old mathematical problem to find Euler bricks, or perfect cuboids. These are orthogonal boxes with $6$ or $7$ integer side lengths, face diagonals, and body diagonals. A perfect cuboid ($7$ integer lengths) has not yet been found, nor excluded, but there are almost perfect cuboids with $6$ integer lengths. Among them the cuboid with side lengths $104$, $153$, $672$, face diagonals $\sqrt{104^2+153^2}=185$, $\sqrt{104^2+672^2}= 680$, $\sqrt{474993}$, and body diagonal $697$.

Since there is no requirement about the base diagonal in the pyramid we can use this cuboid to construct a pyramid of the required kind: Take a base rectangle with side lengths $2\cdot153=306$ and $2\cdot672=1344$. In the center of this rectangle erect a height $104$ to the peak of the pyramid. The slant heights then are $185$ and $680$, and the edges going to the peak have length $\sqrt{153^2+672^2+104^2}=697$.