Calculating cutting angles from Women in Woodworking

At times, simple questions trigger rather complex, but necessary, answers. This one on the Women in Woodworking web site did just that. - Editor

"What is the correct angle cut to get a hexagon without gaps?" - Deb

First came a very concise, and correct, answer. - Editor

"360 degrees divided by 6 sides = 60 divided by 2 (the 2 pieces that come together) = 30 degree cuts." - Joyce

Then came a puzzling entry suggesting a different answer, and some responses to it. - Editor

"The cut for a Hex would be 31.62 degrees." - Joe

"Why the extra 1.62 degrees?" - Johanna

"It's not extra. I've used 30 and it's worked out fine, but according to a chart I have, the correct angle for a hex would be 31.62. Also, if you own a DeWalt electric miter saw, you'll also see the stop for a hex at 31.62. Why, I must confess, I don't know." - Joe

Well, someone knew, and was able to clear up the confusion, and add to it at the same time. - Editor

"Joe, the preset 31.62° indent on the miter gauge of chop saws is to facilitate cutting crown molding laid flat on its back on the saw table. It has nothing to do with cutting miters for hexagons, which really is 30°. U.S. crown moldings are often made with the configuration of a 38° spring from the wall, meaning a complementary rise of 52° from the ceiling. With a bit of calculating, the requisite dihedral angle of the crown molding parts meeting at the corner can be worked out along with the miter angle of each part resulting in a compound cut. For a molding to meet square in a theoretically perfect 90° room corner that rises from the ceiling at 52°, the dihedral angle is 112.27°. To cut the appropriate bevel the saw blade is tilted to cut half this; i.e., 56.14°, or more usually its complement of 33.86°. 33.86° is usually marked on the bevel protractor at the back of the machine. The requisite miter angle for this cut is 58.38° the complementary angle being 31.62° Chop saw makers know it's not easy for most people to calculate these angles so they have pre-set indents and marks on the protractors and miter gauges." - Sgian

Confused? You're not alone, but read on, and more will be clarified. - Editor

"Those are some calculations! The ‘requisite dihedral angle?' I hope ya'll forgive me but I hadn't a clue what that is." - Deb

"Deb, the term dihedral is used in a couple of circumstances I know of. It's found in the description of wing profiles on airplanes. For we woodworkers it means quite simply 'the intersection of two planes.' Take a simple rectangular box viewed from directly above the corner. The intersection of the sides (dihedral angle) is 90°. To join the sides with a miter, divide the dihedral angle by 2 and cut each piece at 45°. To join the two sides with locking type joint showing a square profile as seen from above you subtract the dihedral angle from 180° (i.e., in this case 180- 90 = 90) and the joinery is therefore executed at 90° to the faces: e.g., a through dovetail, lap dovetail, tongue and housing, biscuits, dowels, etc. Crown molding is akin to a pyramid. If you look directly down on a pyramid from a point perpendicular to the corner it is square. Yet to establish the dihedral angle you need to look down a pyramid from its upper point along the arris of the intersecting sides to the bottom corner and measure the angle the sloping faces of the planes form at the bottom corner. Secondly, to establish the angle of the faces of the pyramid sides as they rise from the ground you don't view them as if you are looking at them from a point perpendicular to the ground. You must view the face from a point perpendicular to that face which will give you the angle the side edges describe to the bottom edge.

In a pyramid where the sides lean in from the vertical at 60° the dihedral angle is 104.48° (with a miter angle of 52.24° to join them), and the angle that the triangular sides edges rise from the base line is 63.43°. To dovetail a pyramid type structure of this type with the angles I detailed, as in a butler's tray, you have to rake the dovetails and the rake is established by the sum 180°- 104.48°= 75.52°. Four, six, eight and other even numbered sided items that have sides that rise vertically from the base are comparatively easy - e.g., picture frames, eight sided boxes and the like - but it gets pretty complex when you have odd numbered sided polyhedrals with their sloping sides." - Sgian

But wait, there's more. - Editor

"Incidentally, most people don't bother trying to work out all those angles.
What most people do is spring the molding up at the requisite angle from the saw table, leave the saw bevel setting at 90°, and just flip the table miter gauge to each 45° setting and chop through. This way you have to imagine that the saw table is the underside of the ceiling and the rear saw fence is the wall. With the common 52/38° crown molding combination the molding springs up from the saw table at 52° towards the rear fence. Most people clamp a straightedge as a stop on the table to set the 52° spring, although many machines now come with extra doodads to help." - Sgian

Hey, there's good stuff in there, even if you have to read it several times to get it all down. - Editor