Think of the cone as a stack of thin circular discs, and use the standard centroid formula for a solid of revolution.
Place the apex at the origin and let x run along the axis toward the base. At a distance x from the apex, the radius of the disc is \(r(x) = \dfrac{d}{2h}x\), so its area is \(A(x) = \pi r(x)^2 = \dfrac{\pi d^2}{4h^2}x^2\).
The centroid distance from the apex is \(\bar{x} = \dfrac{\int_0^h x\,A(x)\,dx}{\int_0^h A(x)\,dx} = \dfrac{\int_0^h x^3\,dx}{\int_0^h x^2\,dx} = \dfrac{h^4/4}{h^3/3} = \dfrac{3h}{4}\).
Since the total height is h, the distance from the base is \(h - \dfrac{3h}{4} = \dfrac{h}{4}\).
\[\boxed{\dfrac{h}{4}}\]