Measurements of the grand total and positronium-formation cross sections for positrons scattered by helium within the impact energy range from 10 to 60 eV are presented. All measurements presented here use a high-resolution $(\ensuremath{\sim}70\text{ }\text{meV})$ trap-based pulsed positron beam. Scattering is studied using a high-magnetic field, and absolute measurements of the scattering cross sections are obtained without the need for normalization to other cross sections. We also present single center, convergent close coupling calculations of the total cross section. A detailed study of the cross section to investigate the possibility of scattering resonances and channel coupling has been made. Comparisons with previous cross-section measurements and theoretical calculations are also included.