In this work we are reporting on the measurement of the proton-air inelastic cross section sigma(inel)(p-air) using the Telescope Array detector. Based on the measurement of the sigma(inel)(p-air), the proton-proton cross section sigma(p-p) value is also determined at root s = 95(-8)(+5) TeV. Detecting cosmic ray events at ultrahigh energies with the Telescope Array enables us to study this fundamental parameter that we are otherwise unable to access with particle accelerators. The data used in this report are the hybrid events observed by the Middle Drum fluorescence detector together with the surface array detector collected over five years. The value of the sigma(inel)(p-air) is found to be equal to 567.0 +/- 70.5[Stat](-25)(+29)[Sys] mb. The total proton-proton cross section is subsequently inferred from Glauber formalism and the Block, Halzen and Stanev QCD inspired fit and is found to be equal to 170(-44)(+48)[Stat](-17)(+19)[Sys] mb.