Do protons have a shape? They are often assumed to be balls and spheres in high school textbooks and exams and I had never questioned this until somebody complained to me that modern physicists are obsessed with balls.
Do protons have a shape and if so, what experiments have been done to determine their shape?
 A: Lots of experiments to determine the shape of the proton have been performed, beginning in the 1950's. The answer to "what is the shape of a proton" depends on what energy you use to probe it, as follows.
at low energies, a particle (for example, an electron) fired at a proton bounces off it as if it were a tiny sphere with a finite and measurable diameter. But as you put more energy into the projectile particle, it begins to force its way into the interior of that sphere, where it then encounters a very different world.
By measuring the angles at which the electrons are scattered off of the interior of the proton, the experimenters could determine that there were little doots inside the proton that were point-like, carried fractional electric charges, and that there were three of them.
These experiments were called deep inelastic scattering experiments and the doots thus revealed were named quarks. See Riordan's book The Hunting Of The Quark for an excellent, nonmathematical history of this topic.
A: Yes and no. They have a shape, but it isn't like the shape of a classical object. @Ghoster's link, Shapes of the Proton, describes this. I would like to add to that.
A proton is not a fundamental particle. It is comprised of quarks, a bit like an atom is comprised of electrons around a nucleus.
An electron is something like a point particle. That is if it has a size, it is smaller than any experiment so far can measure.
When an electron is bound to a nucleus, an orbital is the region where the electron is. People talk about the size and shape of atomic orbitals all the time. In a sense, the electron takes on the shape of the orbital.
It is wrong to think of electrons as point particles orbiting around the nucleus, staying inside the orbitals. First, orbitals are eigenstates of the Hamiltonian. They do not change with time. Electrons doe not move back and forth from one side of the nucleus to the other.
Second, electrons don't have a position like a point particle. Electrons are something like classical waves. When a wave hits the beach, it is spread out. It doesn't have a point-like position. But it is misleading to think of them as classical waves. They are not some sort of fluid that fills the orbital. It is also wrong to think of them as a river flowing around the nucleus. They are not spread out in the same sense as anything classical.
A classical fluid has parts. One part is here, and another there. Each part has a definite size, shape, and position. An electron has no parts. If an energetic charged particle comes flying by an atom, it might knock the electron out of the atom, or kick it into a higher energy orbital, or it might miss entirely. It never kicks half an electron out or kicks half an electron into another orbital. No matter how small or well-localized it is, the particle always interacts with an entire electron or no electron.
Also, we never see one part of an electron repel another part of itself. The motion of a free electron is guided by the Schrodinger equation. That is, it is attracted to regions of low energy, and tends to continue moving in the direction it is going as required by momentum. There is no term in the Hamiltonian for Coulomb repulsion with itself.
But there is a sense in which this point-like no-parts electron does fill its orbital. The shape of the orbital defines where it has a "presence". If a well-localized charged particle interacts with an electron, the interaction is more likely to occur in the core of the orbital than in the fringes.
This is much like the behavior of the double slit experiment. If a spread out electron encounters a double slit, it goes through both slits. Then it continues on to hit one atom in the screen. It has a spread out presence as it goes through the slits, but it is one part when it hits the screen.
A proton contains 3 point-like quarks that attract each other. There is no nucleus, so there are no orbitals like an atom has. But the quarks do stay in a small region of space. And this region is what we mean by the size and shape of the proton.
A: I suggest looking into density field theory. But overall a proton acts as a single electron doner. Ie it almost always loses its electron in bonding and takes on a partial positive charge or total positive in the case of ionization. The orbital shape will "lean" towards wherever its electron may be attracted be it bond or anion. Think how the water around earth bends to the moon. Ie even without an electron, the orbitals shape will be a hybridization of some s and some p, perhaps d if the necessary nuclei are present to divide space and time in such an manner.. see orbital theory and learn to draw every orbital of water :)
