For an n-dimensional simplex n and any point D in spherical space S n (1), we establish an inequality for edge lengths of n and distances from point D to faces of n , and from this we obtain some inequalities for the edge lengths and the in-radius of the simplex n . Besides, we establish some inequalities for the edge lengths and altitudes of a spherical simplex, and we establish inequalities for the edge lengths and circumradius of n . MSC: 51M10; 52A20; 51M20