The internal gravity and self-gravitational energy of a comet, asteroid, or small moon have applications to their geophysics, including their formation, evolution, cratering, and disruption, the stresses and strains inside such objects, sample return, eventual asteroid mining, and planetary defense strategies for potentially hazardous objects. This paper describes the relation of an object’s self-energy to its collisional disruption energy, and shows how to determine an object’s self-energy from its internal gravitational potential.Any solid object can be approximated to any desired accuracy by a polyhedron of sufficient complexity. An analytic formula is known for the gravitational potential of any homogeneous polyhedron, but it is widely believed that this formula applies only on the surface or outside of the object. Here we show instead that this formula applies equally well inside the object.We have used these formulae to develop a numerical code which evaluates the self-energy of any homogeneous polyhedron, along with the gravitational potential and attraction both inside and outside of the object, as well as the slope of its surface. Then we use our code to find the internal, external, and surface gravitational fields of the Platonic solids, asteroid (216) Kleopatra, and comet 67P/Churyumov–Gerasimenko, as well as their surface slopes and their self-gravitational energies. We also present simple spherical, ellipsoidal, cuboidal, and duplex models of Kleopatra and comet 67P, and show how to generalize our methods to inhomogeneous objects and magnetic fields.At present, only the self-energies of spheres, ellipsoids, and cuboids (boxes) are known analytically (or semi-analytically). The Supplementary Material contours the central potential and self-energy of homogeneous ellipsoids and cuboids of all aspect ratios, and also analytically the self-gravitational energy of a “duplex” consisting of two coupled spheres. The duplex is a good model for “contact binary” comets and asteroids; in fact, most comets seem to be bilobate, and might be described better as “dirty snowmen” than as “dirty snowballs”.