Extruding from square to round
I would use a lofted boss/base for this if you want to do it the simple, solid way. Lofted surface would be more appropriate for a more complex surface model, but here's how you loft a solid boss.
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Step 1:
Select a plane and create a sketch. In this case I am using the front plane. Draw/dimension your circle. Mine here is 5" diameter, centered at the origin. Exit sketch.
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Step 2:
Select the front plane again and click Insert-->Reference Geometry-->Plane. In the dialog box, click on distance and enter the distance that you want between your circle and square. Here I've chosen 6". Click OK.
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Step 3:
Select your new plane and create a sketch. Draw/dimension your square. Here I've made it 8" square, centered at the origin again. Exit sketch.
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Step 4:
Select Insert-->Boss/Base-->Loft. Right click an empty space on your model window and select Selection Manager.
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Step 5:
A small dialog box will open up. Hit Select Closed Loop.
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Step 6:
Select your square and click OK in the Selection Manager box.
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Step 7:
Select your circle. Click OK in the Selection Manager
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Step 8:
You will now see your loft preview (if you have Show Preview selected in the Loft dialog box). Hit Cancel in the Selection Manager.
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Step 9:
Click OK for the Loft.
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Step 10:
And there it is!