So I know that the reason why planets and other celestial bodies are round is due to gravity, and I know any flat/disk world would buckle under gravitational pressure to reform itself into a sphere. But my question is: what would you have to do in order to sustain a disk shape? How much energy or force (relative to the central mass, and radius of the disk) would you need to exert on the edges in order to retain its shape - Is there a mathematical formula for this? And could this ever occur naturally (denser masses at the edges, for instance), or would it have to be man-made?

I don't have a formula, but qualitatively, the faster it is rotating, the flatter it will be.

Magic

So I know that the reason why planets and other celestial bodies are round is due to gravity, and I know any flat/disk world would buckle under gravitational pressure to reform itself into a sphere. But my question is: what would you have to do in order to sustain a disk shape? How much energy or force (relative to the central mass, and radius of the disk) would you need to exert on the edges in order to retain its shape - Is there a mathematical formula for this? And could this ever occur naturally (denser masses at the edges, for instance), or would it have to be man-made?

According to some sources, a torus planet could form naturally. Not sure how 'flat' they could be though.

See this thread: >> Torus planets