I'm trying to cut a hole about 7" diameter in a corian counter top to use for trash. How do I get a finished (smooth) edge like the edge of the countertop itself?

MiK=meek

as smooth don't know but you may do the following
sand it with a small piece of sanding paper then apply the siding the little thing that look just like your counter top but extremely thin

ballengerb1

You can treat Corian just like a hardwood. I would cut the hole using a zip saw and a circle jig followed with a router to creat a neat rim.

kidolph

Even after you use the router it is best to sand with 220 or 400 grit sandpaper to eliminate all trace of tooth marks. A tooth mark in Corian is like a scratch in glass. It is a stress riser, a place for a crack to start.

heaphy

I've done this sort of thing many times before. My method is to make a template out of half inch hardwood plywood or MDF. Cut an accurate, exact hole the size you want, and sand it smooth. Place the template on the countertop in the correct position, and draw a pencil line. Now, cut a rough, undersized hole inside the pencil line using a jigsaw with a fine woodworking blade, or a Rotozip with the zip bit that comes in a blue package for cutting wood. Place the template in position using bar clamps and/or hot melt glue. Use a router and a three-quarter inch straight cut bit with a ball bearing pilot guide above. Trim the edge smooth and precise. Use a quarter inch radius roundover bit to smooth the edge, and then hand sand as Ken Dolph described. You can use a fixed template guide on your router and an ordinary straight bit as well - you just need to oversize your template to compensate for the offset dimension. If you used hot melt glue to secure the template, a mist of denatured alcohol will help release it without damage.