Dihedrals and Equal Sided Polygons.

This is the case of all equal sided or "regular" polygons where the vertex angle that is formed is not

90°. This example shows a five sided polygon with an apex directly above its center point. Even though the steps are similar to those of calculating a "Pyramid" there are one or two differences.

The First being that you will need to calculate the vertex angle. Since it's not a given 90° you just can't plug it in. You might want to make/keep a chart of the different vertex angles for the different number of sides.

In the example that I will be using there are 5 sides . The base length of one side is 12". The height of the apex above center is 24". I'll start by first showing how to obtain the vertex angle for this polygon.

VA = (180-(360/NS)) /2 VA defines the vertex angle and NS the number of sides.
VA = (180-(360/5)) /2 = (180-72) / 2 = 54°. VA = 54°
PB(Perpendicular Bisector) = 1/2 base * Tan(VA)
PB = 1/2(12) * Tan(54) = 6 * 1.3764 = 8.2583"
This will give us the length of the line that "bisects" the base. This line length is needed to calculate both the side slope and the "miter gauge" angle.
SA = Tan^-1(Height/PB)
Tan^-1(24/8.2583) = 2.9061
Tan^-1(2.9061) = 71.01°
This gives us the "slope Angle"
Once we have the slope angle it's easy to calculate the "bottom cut angle."
BC = 90° - SA = 90° - 71.01° = 18.99°
SL (Slope Length) = PB / Cos (SA)
SL = 8.2583 / Cos (71.01)
SL = 8.2583 / .3254 = 25.3786"
This will now give us all the information that will be needed to calculate the correct "miter gauge" angle.
MGA = Tan^-1(SL / 1/2 base)
MGA = Tan^-1(25.3789 / 6 )
MGA = Tan^-1(4.2297)= 76.698°
With this calculation we now have one of the three settings needed to successfully cut this dihedral angle.
HR (Hip Run) = 1/2base / Cos (VA)
HP = 1/2 (12) / Cos (54)
HP = 6 / .5877 = 10.2078
Now with this information we can calculate the "Hip Angle", this is the angle that the "miter pieces" make with a level line.
HA (Hip Angle) = Tan^-1(Height/HR)
HA = Tan^-1(24/10.2078)
HA = Tan^-1(2.3511) = 66.9587°
HL (Hip Length) = HR / Cos (HA)
HL = 10.2078 / .3914 = 26.081"
We need just a little bit more information to calculate the dihedral angle. Here's how to calculate them.
C = 1/2 base * Tan(HA)
C = 6 * Tan(66.9587)
C = 14.1068
D = C * Cos(HA)
D = 14.1068 * Cos(66.9587)
D = 5.5213
E = D / PB
E = 5.5213 / 8.2583
E = .6686
F = Tan^-1 (E)
F = Tan^-1 (.6686) = 33.766°
This is how to now calculate the dihedral angle and its miter.
DA = (90° - 33.766°) * 2 = 112.468°
MDA = 33.766°
Now all you need to do is tilt your table saw blade to 33.766° and send you work piece through at 76.698° relative to the blade and you will cut a perfect miter joint.

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