1.73 femtometer!
A neutron is a particle with no electric charge, found with protons in the nucleus of almost every atom. It is about 1.7 femtometers across, measured as twice its magnetic radius, which describes how its magnetism is spread out. A femtometer is a millionth of a billionth of a meter (10−15 meters).
A neutron has no hard surface. It is made of three smaller particles called quarks, one up quark and two down quarks, held together by the strong force, the same force that binds neutrons and protons into a nucleus. The up quark has a positive charge and the down quarks have negative charges, and together they cancel out.
A proton's size is measured from how its electric charge is spread out, but that does not work for a neutron. Its positive and negative charges are spread out differently from each other, so the usual measure of charge size comes out as a negative number, which cannot be a width. Physicists use the neutron's magnetism instead.
A neutron acts as a tiny magnet. The Particle Data Group, which collects the world's particle measurements, gives the average radius of that magnetism as 0.864 femtometers, known to about 1%. Scattering experiments also show that neutrons and protons are about the same size.
In 1930, Walther Bothe and Herbert Becker fired alpha particles, which are helium nuclei, at the metal beryllium. It gave off a strong, penetrating radiation, which some physicists took to be high-energy radiation of the same kind as light. In 1932, James Chadwick, working at the Cavendish Laboratory in Cambridge, England, showed that it was made of neutral particles with about the same mass as a proton. Ernest Rutherford had suggested that such particles might exist in nuclei, and they were named neutrons.
Because a neutron has no charge, the positive charge of a nucleus does not push it away. So neutrons can reach and split even the heaviest nuclei, and Chadwick's discovery led the way to splitting uranium. He received the 1935 Nobel Prize in Physics for it.
Outside a nucleus, a neutron does not last long. On average it survives about 15 minutes (878 seconds) before it decays into a proton, an electron and a particle called an antineutrino. A neutron is slightly heavier than a proton, by about 0.14%.