[mjlee]

But manhole covers are always bigger than the hole. Otherwise, they'd fall in at installation and the problem would be immediately obvious.

The point here is more that a circle has the same diameter regardless of orientation. There's no way to rotate it to make it fit through a hole it couldn't before, like you can do with a rectangle.

[jpindar]

This is also true of a triangle, and indeed there are triangular manholes.

[sobani]

Aren't the altitudes/heights of a triangle shorter than its sides?

The height of an equilateral triangle is about 87% of the length of one of its sides.

[jpindar]

True. Whether they fit through the hole depends on how thick they are.