In this study, the transition Reynolds number of compressible axi-symmetric sharp cone boundary layer is predicted by using a linear stability theory and the -method. The compressible linear stability equation for sharp cone boundary layer was derived from the governing equations on the body-intrinsic axi-symmetric coordinate system. The numerical analysis code for the stability equation was developed based on a second-order accurate finite-difference method. Stability characteristics and amplification rate of two-dimensional second mode disturbance for the sharp cone boundary layer were calculated from the analysis code and the numerical code was validated by comparing the results with experimental data. Transition prediction was performed by application of the -method with N=10. From comparison with wind tunnel experiments and flight tests data, capability of the transition prediction of this study is confirmed for the sharp cone boundary layers which have an edge Mach number between 4 and 8. In addition, effect of wall cooling on the stability of disturbance in the boundary layer and transition position is investigated.