Hello,
G center of gravity of a triangle is the intersections of the "médianes" (line passing through a vertice and the middle of the side opposed sorry i don't know the name in English) ans is at 2/3 from each vertice.
A cone is a triangle in rotation around a line (the axis). Uisng Theorem of Guldin, G is on that axis
Altitude of G=1/3 h.