Accurate Miters
Did you ever wonder why it's so hard to cut accurate miters? How can you avoid having a gap after all the pieces are assembled?
In the example above, it would seem like the gap is pretty big and should be pretty easy to eliminate. But, that 1/8" gap is the result of each cut having only a very small error. When all the pieces are put together, the error of each cut accumulates. In an octagon, there are a total of 16 cuts to be made (two cuts on each of eight pieces). So, the error in each cut is 1/16 the size of the gap.
Accuracy Required
If the accuracy of miter cuts in measured in terms of degrees, then the amount of acceptable error depends on the length and width of each member. Longer joints (from wider boards) will require higher angular accuracy than shorter joints. For example, if each joint is 2" long, then a 1/64" gap results from a total error of about 0.45 degrees(arc sin (1/64) / 2). For the octagon, the error in each cut will need to be 0.028 degrees or less (16 cuts in all). For a 3" joint the total error goes to about 0.30 degrees and each cut will have to be accurate to 0.019 degrees.
Accuracy Measurements
Rather than determine how long each joint is and convert the gap width to degrees, it's much easier to measure angular accuracy in terms of how much gap (or deviation) exists between the perfect (ideal) angle and the sample being measured. This is exactly how manufacturers specify the accuracy of squares and angle blocks. Then it's easy to predict and measure how accurately each cut must be. For the example shown in the photo above, the 1/8" gap is produced by having 1/128" error in each cut (1/8 divided by 16). If 1/64" gap is needed for the octagon, then each cut must be accurate to 1/1024" (1/64 divided by 16). In other words, there must be slightly less than one thousandth of an inch error for each cut in order to produce an octagonal frame with a total error of less than 1/64".
No wonder it's so difficult to obtain accurate miters! The old adage that says working wood to within 1/64" is "good enough" just doesn't apply to cutting miters. Such an error would produce a 1/4" gap in the example above! Even a simple four sided frame would end up with a 1/8" gap.
Method for Success
How is it done? There are basically three methods you can use to achieve accurate miters. But, before you launch into one of them you need to know about possible sources of error and how to avoid them. Please read External Influences on the Accuracy of Results in the Technical Documentation Section of the web site.
1. Simple Trial and Error
Just like it sounds, you give it a try and see what happens. For the octagon, it means cutting all eight pieces and seeing how they fit. You make an adjustment to your machine based on examination of the results and try again. Given enough time, patience, and scrap wood, it's actually possible to obtain good results with this method. But the word "simple" is a bit misleading. This is definitely a very difficult task. You'll be called upon to judge the gap and determine how much adjustment is necessary. In the end, you'll be trying to make adjustments to within thousandths of an inch and these are extremely difficult to do by eye. Most people who use this method generally settle for a larger gap than they originally wanted and resort to methods for hiding that error (like trying to even it out among all the joints). Many just give up and avoid miters altogether.
2. Trial and Error with Accurate Measurement
If you could accurately measure the results of a test cut, then you can save yourself a lot of time and scrap wood. Precise adjustments to your machine will still require significant judgment. And, devices that can measure angles this accurately are fairly expensive. The common protractors available from woodworking dealers is completely inadequate. The typical machinists protractor graduated to five minutes of arc generally starts at about $140. The big advantage of this method is that you avoid the need to cut out all the parts for every trial. Another advantage comes from knowing exactly what your machine setting produces. When you measure the result directly, there's no question about what to expect.
3. Accurate Machine Setup
In order to avoid test cuts altogether, you have to know what your machine setting will produce before the first cut is made. To do this your machine must be carefully aligned. You must also be able to make accurate measurements of the machine settings. This is the approach that machinists use to produce reliable and repeatable results. Generally their machines come equipped with accurate measurement devices. Unfortunately, the adjustments on most woodworking machines are very crude by comparison. So, an accurate alignment/setup tool is needed so that machine settings can be accurately measured and adjusted. Such a tool generally starts at about $70. A growing number of woodworkers are coming to appreciate this approach and many manufacturers are now producing aftermarket accessories that help facilitate it. This is the method that I advocate.
Last revised: January 21, 2006.