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In theory, at least, you could miter crown moulding when two pieces meet in a corner. In fact, however, it would be a disaster. Crown only touches the wall at two points--the top and the bottom--so the vast majority of the joint is floating in midair, making it impossible to get a tight seam. Coping the corner pieces solves the problem. Here's how it works: One moulding runs into the corner of the room, where it meets the wall with a simple butt joint. It supports the second moulding, the end of which is cut to fit against the first molding like a piece of a jigsaw puzzle.
Putting the first moulding in place is a matter of cutting a butt joint and nailing the moulding in place. Cutting the second moulding to fit against it begins with a miter cut that makes it easy to see the shape you need to cut. You then cut along the shape with an inexpensive handsaw called a coping saw. Make sure you've got a fine blade in it (check it against a pack of replacement blades if you're not sure.) You'll also need a stable work surface one to three feet above the ground--a toolbox or sawhorses will do, but a platform like the one shown here supports more of the moulding and is easier on your back.
Despite appearances, this is not a hard joint to learn, but practice helps. Buy an extra piece of moulding before you start, and cut a few practice joints. It will give you confidence, and you'll get to throw out your mistakes instead of hanging them on the wall for all the world to see. The next few pages show you how to cut the end of one molding to fit against the other.

Lay out and miter the cope.
Nail the first piece of
moulding in place. Hold the second piece temporarily in place, and lay out the
direction of the miter by drawing a line that starts at the bottom of the
moulding and angles up and away from the corner. The line is only a guide and
doesn't need to be drawn at any particular angle. Put the moulding on the
miter saw, and set the saw to cut a 45-degree miter that angles in the general
direction of the line you drew, and cut a miter.

Begin at the flat along the bottom edge of the moulding.
Put
the saw blade on the end grain just below the surface of the moulding, as
shown. (If the moulding is primed, you'll lay the saw just a hair below the
lower suface of the paint.) Make a few passes with the saw, making a shallow
groove along the entire length of the flat.

Change the angle of the saw.
Once you've made a
shallow cut into the flat, lift the saw handle and begin cutting along the
bottom corner of the moulding. Cut from the corner to the point where the cove
begins, following the groove you cut in step two.

Cut in from the back.
Back the saw out of the cut
you've made. Start a new cut from the back of the moulding. Cut up towards the
middle of the cove, and then follow the shape of the cove until you reach the
cut you finished in step three. Remove the waste.

Cut out the upper section of the cove.
Turn the saw
around, and cut along the profile of the upper section of the cove. Stop
cutting when you reach the fillet.

Cut out the fillet.
Come in from the back of the
moulding, and cut along the fillet until you reach the cut you made at the top
of the cove.

Cut along the S-curve.
Start at the fillet and cut
along the S-curve until you reach the flat at the top of the moulding.

Cut out the flat along the top of the moulding.
Put
the saw on the end grain just below the surface of the molding. Make a few
shallow passes, and then raise the handle so that the saw is cutting deepest
at what will be the top edge of the molding. Continue cutting until the saw
meets the cut along the S-curve and the waste falls away.
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