Suppose the cylindrical symmetry of the problem to note that the center of mass must lie along the z axis (x = y = 0).
If the uniform density of the cone is ρ, then first compute the mass of the cone. If we slice the cone into circular disks of area pi r^2 and height dz, the mass is given by the integral:
M=∫ρdV=ρ∫pi r2dz from zero to h.
M=ρ∫pi a2(1−z/h)2dz from zero to h = pi a2 ρ∫ (1−2z/h+z2/h2)dz from zero to h = 1/3pi a2 h ρ from zero to h.
z=1/M∫ρzdV
z=1/4h