Let's compute the centroid of the semi-circle above the x axis. By symmetry, the point lies on the y axis; we only need find the y coordinate. Switch to polar coordinates and integrate y, or r×sin(θ). Bring in the extra r as required, and obtain an integral of 2/3. Now divide by the area of the shape, which is π/2. The centroid is at 4 over 3π, approximately 0.4244. This is less than half the radius, which makes sense, since most of the circle is near the x axis.
If a cone has base b and height h, stand it up on its point, like an ice cream cone. Its volume is bh/3, found by integrating b(z/h)2 as z runs from 0 to h. The centroid is on the axis; we only need find the z coordinate. Integrate bz(z/h)2 to get bh2/4. Divide by volume and get ¾h. Therefore the centroid of a cone, or pyramid, or any other shape whose dimensions decrease linearly from base to apex, is on axis, ¼ of the way from base to apex.