  In this work we are reporting on the measurement of the proton-air inelastic
cross section $\sigma^{\rm inel}_{\rm p-air}$ using the Telescope Array (TA)
detector. Based on the measurement of the $\sigma^{\rm inel}_{\rm p-air}$ the
proton-proton cross section $\sigma_{\rm p-p}$ value is also determined at
$\sqrt{s} = 95_{-8}^{+5}$ TeV. Detecting cosmic ray events at ultra high
energies with 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 is the hybrid events observed by the Middle Drum
fluorescence detector together with the surface array detector collected over
five years. The value of the $\sigma^{\rm inel}_{\rm p-air}$ is found to be
equal to $567.0 \pm 70.5 [{\rm Stat.}] ^{+29}_{-25} [{\rm Sys.}]$ mb. The total
proton-proton cross section is subsequently inferred from Glauber Formalism and
Block, Halzen and Stanev QCD inspired fit and is found to be equal to
$170_{-44}^{+48} [{\rm Stat.}] _{-17}^{+19} [{\rm Sys.}] $mb.
