Accurate knowledge of the charge and Zemach radii of the proton is essential, not only for understanding its structure but also as input for tests of bound-state quantum electrodynamics and its predictions for the energy levels of hydrogen. These radii may be extracted from the laser spectroscopy of muonic hydrogen (μp, that is, a proton orbited by a muon). We measured the $2{\mathrm{S}}_{1/2}^{\mathrm{F}=0}-2{\mathrm{P}}_{3/2}^{\mathrm{F}=1}$ transition frequency in μp to be 54611.16(1.05) gigahertz (numbers in parentheses indicate one standard deviation of uncertainty) and reevaluated the $2{\mathrm{S}}_{1/2}^{\mathrm{F}=1}-2{\mathrm{P}}_{3/2}^{\mathrm{F}=1}$ transition frequency, yielding 49881.35(65) gigahertz. From the measurements, we determined the Zemach radius, r Z = 1.082(37) femtometers, and the magnetic radius, r M = 0.87(6) femtometer, of the proton. We also extracted the charge radius, r E = 0.84087(39) femtometer, with an order of magnitude more precision than the 2010-CODATA value and at 7σ variance with respect to it, thus reinforcing the proton radius puzzle.