In both fiction and non-fiction worlds, that attempt to keep a sense of realism, planets are spherical.
Is it possible for a planet to not be shaped like a sphere?
If such a planet is possible could it sustain life? For example, having a proper magnetic field, atmosphere, etc?
It'd be possible for one to exist for a time, but a naturally occurring, non-spheroid planet would be incredibly unlikely. More on that at the end.
It's an easy thing to imagine of course, but that's because we think of things like cube shaped rocks that occur naturally and think 'why not?'. The problem is that objects on a planetary scale don't behave the same as boulder or even small moon sized things. Anything above 200-300km radius squishes into a spherical shape.
So, you can have a space potato that's not a sphere, but it's technically not a planet. It could have an atmosphere at any size, though it's less likely to hold on to it if there are other large masses nearby or in high solar winds.
To have something the size of a planet but not a sphere you'd have to find a planet that has had a terrible accident. If you crash a large moon into a planet you might be able to horribly deform the planet and give it some new moons. Assuming the remaining mass is large enough it could still be defined as a planet (massive enough to collapse into a sphere) but non-spherical. The non-spherical shape would be temporary, hundreds of thousands of years perhaps. As it's just been destroyed it's unlikely to have life though.
Edit:
What if Earth were a cube?
There is an interesting article on that here. Basically the corners and edges of the cube would be like massive mountain ranges, as they would be at an angle with respect to the vector of gravity (except in the center of the faces). I wrote a more detailed answer about such a world here.
The oceans would move to the centers of the cube faces. Basically, the Earth would look very different. Likely still inhabitable, assuming whatever magic formed it into a cube kept the the shape, otherwise we'd probably all be killed by the planet shaking earthquakes.
Donut-shaped planets are indeed possible!
The article cites a paper which concludes that certain kinds of toroidal planets are at least internally stable, although not likely and possibly would get ruined by any external interference.
It's true that gravity will make a randomly-shaped blob of matter coalesce into a sphere over time, but there's no reason that it has to do so with a torus. Matter is attracted most strongly to the closest matter.
Yes, we live on one :-)
Strictly speaking, the Earth is not a sphere, it's an oblate spheroid. Its rotation makes it bulge somewhat, so the equatorial radius is ~30 km greater than the polar. A faster rotation would make a planet even more oblate: Saturn's polar & equatorial radii differ by almost 10%: https://en.wikipedia.org/wiki/Saturn#Physical_characteristics because the centrifugal force from rotation counterbalances the gravity.
Cubes, donuts, &c couldn't exist, though, at least if they're composed of normal matter. It's a matter of strength of materials. The corners of a cube would behave like very tall mountains, and collapse under their own weight.
The sphere shape is entirely based on gravity. That is actually one of the parts of the definition of a planet. That is is large enough that it's gravity will make it a sphere. The objects that are not spheres are much smaller or they would have to be artificially created and maintained. (unless shortly after a large event that distorted the planet, like a moon colliding into a planet, but in time it will round itself out again.
Hard SF writer Hal Clement wrote a delightful book in 1953, Mission of Gravity, which introduced a very physically correct non-spheroidal world, Mesklin. Mesklin, a high-mass planet subject to extreme rotational stress, is shaped like a thin disk with a central bulge. From the Wikipedia article:
"Clement decided, since its mass was 16 times that of Jupiter, Mesklin would have an extremely large angular frequency to partly counter its gravity in order to allow humans to visit part of it. He wanted the equatorial gravity to be 3 g, so he determined the period necessary to make this occur: each Mesklin day is 17.75 minutes long given that the planet rotates approximately 20 degrees a minute.
As a result of this extremely large rate of spin, Mesklin is not even slightly spherical; it has a large equatorial bulge. Mesklin's equatorial diameter is 48,000 miles (77,250 km), while from pole-to-pole along its axis of rotation it is 19,740 miles (31,770 km). Then Clement attempted to calculate the polar gravity, finding it surprisingly difficult. He admits, "To be perfectly frank, I don't know the exact value of the polar gravity; the planet is so oblate that the usual rule of spheres... would not even be a good approximation..." "Whirligig World" reports his initial calculations of the pole gravity to be 655 g; the dust jacket of Heavy Planet reports it as 700 g. A later program created by Clement computed it as 275 g"
This is a much more extreme example of the deformation of Saturn described in jamesqf's answer. Possibly not what you're looking for, but it makes a nice illustration of just how plastic large planets are under the force of gravity and the apparent centrifugal force of spin.
It's not possible unless the planet is made of unobtainum.
If you had, for example, a cube-shaped planet with a liquid center, you would have a huge amount of pressure on the liquid interior of the planet applied by the corners of the cube, which would effectively function like gigantic mountains. This pressure wouldn't exist on the faces of the cube, so the internal pressure would push them outwards, likely resulting in the formation of some absurdly huge volcanoes as the liquid interior of the planet, under the pressure applied by the corner mountains, burst through cracks in the surface of the low-pressure faces of the cube. The corners would sink down and the flat portions would fill up with lava, which would slowly cool into the shape of a sphere.
What about some planet in orbit around a very heavy black hole?
There could be some situations where the stable state of such a system included a non-spherical planet.
Also, you could get some interesting relativistic time effects, like in Interstellar.
One of the defining traits of a planet according to the IAU is that it be in hydrostatic equilibrium. That means it can't be holding its shape against its own gravity which results in an oblate spheroid with at most fairly minor lumps.
If you made a planetary scale structure out of super string unobtanium (Rather than something as squishy and flexible as rock) that was not in hydrostatic equilibrium, it technically wouldn't be a planet. Ring worlds, orbitals, and the like are not planets.
Unlike the "clear its neighbourhood" requirement (the thing that got Pluto reclassified), no one really disputes this part of the definition of "planet".
The spin rate required to get enough oblateness to be significant to anyone other than scientists, engineers, and the like just isn't going to occur in something that fits the definition of a planet naturally. The collapsing dust cloud forming a star and its planets is only going to have so much angular momentum. It's also going to be subject to some crazy tidal forces (which will slow it down)
In theory a torus (doughnut shape) can be in hydrostatic equilibrium if it's spinning fast enough. Even if it could get spinning that fast, and get into the torus shape, it wouldn't be stable in the long term. If it got even slightly out of balance it would tear itself apart.
The rough version of this is: No, a planet is (to the tolerances of a ordinary person) spherical by definition. Anything not spherical is not a planet.
I think that a planet is definitely able to be misshapen and still be a planet! Our very own Earth is not a perfect sphere! It is a bumpy oblate spheroid, though it looks like a marble from space. Earth's poles are squashed, and it is swollen at the equator. Because of the bulge, the distance from Earth's center to sea level is roughly 21 kilometers (13 miles) greater at the equator than at the poles. This was all discovered by Issac Newton. Earth has a bit of plasticity that allows the shape to deform slightly. An oblate spheroid is not perfectly correct either, because of the fact that mass is distributed unevenly within Earth, also causing more deforms.
It is not possible because of gravity, with the spherical shape of our Earth's core, the earth is shaped into a sphere. If there were a donut shaped planet, there would be no place for the core, therefore it wouldn't be classified as a planet. With the hole in the center, the center of gravity wouldn't exist. To be classified as a planet, it has to have a core, and atmosphere, and has to be shaped as a sphere.
If there was a solar system with two sun's and a planet in-between, if the sun's rotated while the planet was, it might keep the planet in a disklike shape. Or split the planet in half. But that chance is still there. It would look like that one planet from Netflixs Voltron.
