Abstract: In this paper, We established three theorems relating to Von Neumann’s sampling technique, and gave some methods to produce random points uniformly in area on certain particular n-dimensional surfaces, such as n-dimensional sphere x₁²+x₂²+…+x_n²=R²,n-dimensional cylinder x₁²+x₂²+…+x_n-1²=R², h₁≤x_n≤h₂ and n-dimensional surfare x_n= (x₁²+x₂²+…+x_n-1²)^1/2, 0≤x_n≤H, etc. And the results are particularly usefull in moute carlo method.