The recent discovery of ultrahigh mobility and large positive magnetoresistance in the topologically non-trivial half-Heusler semimetal LuPtBi provides a unique playground for studying exotic physics and significant perspective for device applications. As an fcc-structured electron-hole-compensated semimetal, LuPtBi theoretically exhibits six symmetrically arranged anisotropic electron Fermi pockets and two nearly-spherical hole pockets, offering the opportunity to explore the physics of Fermi surfaces with simple angle-related magnetotransport properties. In this work, through angle-dependent transverse magnetoresistance measurements, in combination with high-field SdH quantum oscillations, we aimed to map out a Fermi surface with six anisotropic pockets in the high-temperature and low-field regime, and furthermore, identify a possible magnetic field driven Fermi surface change at lower temperatures. Reasons account for the Fermi surface change in LuPtBi are discussed in terms of the field-induced electron evacuation due to Landau quantization.