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Attaching Two Objects

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I have to attach the bottom of a cylinder to the top side (centre) of a cube.  The cylinder has 200 segments on the circumference and I have deleted the cap on bottom side . The cube is just a cube at this time. 


What is the easiest or best way to attach the cylinder to the top of the cube, bearing in mind I need the geometry to be attached and clean? 

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Divide 200 by 4 to get the amount of subdivisions you will need on each side of the cube, so make a cube with that value in all axes, then using Phong break Selection, select and delete the entire 2500 polys that comprise the lower face of the cube where the cylinder will join it. Now Connect Object and Delete both objects, Outline select both boundaries, check the edge count is divisible by 2 (and should say 400 in the HUD if you are showing selected edges there), and stitch n sew the 2 loops together, holding shift to create a new surface. This most easily done in Top orthographic view, starting in the center of one of the cube's sides. Make sense ?




You can add additional edge loops to the new surface to even up polygon density with either Loop Cut or Edge Cut.



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