The Dihedral Angle

By Joseph Fusco

This is the angle that is formed whenever two "planes" meet. To most people who deal with either woodworking or construction, the angle comes into play when you would like to put a "top" on something or when you frame the hip rafter for a hip roof.

The graphic below shows a pyramid shape with a base of 24" x 24" and a height of also 24". By looking at the graphic you can see where this could also illustrate the corner of a "Hip" roof.

The Dihedral angle is the one formed where any of the four sides meet. The most common shape that most woodworkers deal with is the pyramid or where you have 90° corners and equal sides. This is the type that I will be discussing here. The dihedral angle is shown below in orange.

There are a few steps to follow to get the correct angle for this miter and they will work for any 90° corner with equal sides. There are some sites that offer you a quick way to get the angle, but their calculations are based on the slope angle of the face pieces rather then the height of the pyramid. To me it seems you will have much more control of your work piece if you control the height rather then the slope of the "box".

Using those methods you are bound by a most Static model. With the information here you will be able to design any base and height you want.

The Things you'll need to Know.

Basically the there are three angles you'll need to know in order to cut your pieces on the table saw.

The first being the angle to tilt your saw blade to (the dihedral).
The angle to pass the work piece through at, this could be called the "miter gauge" angle. The graphic below shows this angle.
The base angle to cut the bottom of your pieces so that when they are place the piece "stands" at the correct angle.

You will need a scientific calculator or you can use the windows calculator if you like.

Using the above example this is how you determine these two angles.

Knowing that the base length is 24" and the height is also 24" is all the information that you will need to calculate the angles. This is a bit mathematically intense, but there really is no other way to get the answers.

Piece slope angle and base cut angle.
Using the formula Tan^-1( height / 1/2 base) to find the slope angle. . .
Slope Angle = Tan^-1( 24 / 12) = Tan^-1( 2) = 63.43°.
Bottom cut angle = 90° - Slope Angle = 90 - 63.43° = 26.57°.
Using the Pythagorean Theorem we can now calculate the perpendicular length of this piece. PL = sqrt((height)² + (1/2 base)²) = sqrt((24)² + (12)²) = sqrt(576 + 144)
PL = sqrt(720) = 26.83
Knowing the length of this side, we can now calculate the "miter gauge" angle. Using the formula Tan^-1( PL / 1/2 base) = ( 26.83 / 12) = ( 2.2358) = 65.90°

This now gives you the angle to set your miter gauge to get the correct angle on the piece.

The next piece of information that we'll need to find is the angle that is made when the two pieces come together to form the "corner" Once joined, this joint will have an angle that it makes with a "level" line. This angle is illustrated by the green triangle in the graphic to the right. This is how to calculate that angle.

Using the formula Tan^-1( height / (1/2 base x 1.4142))
Pitch Angle = Tan^-1( 24 / (12 x 1.4142)) = (24 / 16.9705) =Tan^-1( 1.4142)
Pitch Angle = Tan^-1(1.4142) = 54.74°

All that is left now is to calculate the dihedral angle, we can easily (well almost) do it from the information we have now.
C = Height * (Cos (54.74°) = 13.854
D = C / (1/2 base * sqrt(2)) = (13.854 / 16.9705) = .8164
Tan^-1( D ) = 39.23°
The dihedral angle is (90° - 39.23°) * 2 or 101.54°
Now to adjust your saw blade tilt to 39.23° and you're ready to cut!

Try the online calculator!

All written, photographic and drawn materials property of and copyrighted by josephfusco.org 1997-2008.

No materials may be used in any way without the written permission of Joseph Fusco

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