The compound miter saw, like the name suggests is more than a miter saw, but allows two different angles to be cut with one swipe of the blade. It is easy to use, once you understand the combination cuts it can make.
The most typical angle you may want to cut is a forty five degree angle, with the blade held perpendicular to the stock. Common usage include for example, framing a window, or a door frame, where you want your frame pattern to continue up, over, and down around these typical woodworking obstacles. You may also set the blade to cut anywhere in the range of 45 to the left, all the way through to a 90 from the fence, to 45 to the right. When learning how to use this machine, you can guarantee mistakes will be made, but have patience's. Learning the consequences of cutting at different angles will soon become apparent as you learn stock of the same width require the same angle (45) to match up exactly, especially if there is a pattern involved. You will find that the varying other angles available are, most often used for matching up your stock to an existing angle, a ceiling/roof line, for example.
When you're trimming a long length, with stock that is typically shorter than the wall, chair molding for instance, you will find it is easier to cut successive pieces of short (8'), or shorter, lengths, and match them up, using a level line as a guide to make a straight run. But if you simply butt joint them together, anything short of a perfect cut will yield obvious joints that are anything but professional looking. Matching up opposite perpendicular 45 angles will work to hide imperfections, but leave noticeable 45 cuts down the length. A better solution is to leave the blade at 90 to the fence, but lean the blade over 45 instead, and allow the lengths to butt together with these complimentary angles, the pattern will match exactly, because of the 90 angle from the fence, and the imperfections of the cut will be hidden because the angle of the cut will be tight up against the wall where nobody but the keenest (professional) eye will be able to detect that there is a joint there at all.
The use of the compound feature of this saw is most often used by the professional carpenter. When cutting rafters for a roof, most cuts will be will be made with the blade perpendicular to the stock at an angle determined, according to the calculations made with a carpenter square. But when it comes to the hips and the valleys you will be confronted with two angles that need to be cut on the same end of the rafter, a compound angle! One angle is the angle of the rafter to the ridge (the rise), which must be calculated exactly using your carpenters square, and then the second angle, which is the face of the rafter to the same ridge, (typically leaning 45 from perpendicular on your saw) also calculated with a carpenters square. This is a great time saving tool, when you know how, and when to use it.
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