Every woodworker, carpenter and architect has encountered situations that called for designing and cutting materials at compound angles. Crown molding is only one of many applications. This Part 5 goes into more detail and provides you with some theory and ways to solve the problem of calculating the set up for your saw. I will do this by first going through specific examples.

1. Multi sided boxes with slanted sides:

I have made planter boxes such as this "shadow" box that holds a small flowerpot for my wife. It was made out of 2x6 redwood and has a mirror in the back to reflect light. I made this several years ago, and to set the miter and bevel angles I relied on some tables that I found in a wood working book.
Now I can use my spreadsheet for cutting compound angles. See Spreadsheets above.

Note: There were some errors in the original pages. The miter and bevel angle calculations are now corrected. See the spreadsheet for the new formulas. I have also deleted the 'overall' box dimensions.

To verify the new formulas I made the following five sided 'box' with a sloping side of 60 degrees. I want to thank Professor Frank J. Franczak of the University of Wisconsin for pointing out the orginal errors.

Planter boxes with sloping sides that are steeper can also be made this way. But don't limit yourself to six sides. Let's design one and put a sketch together:

To do a decent job and have the edges mate and make a flush joint, you have to cut the parts at compound angles. Select the right material for the project and let's see how we need to make the cuts. 

Using the spreadsheet for compound angle cutting, calculate M (miter angle) and B (bevel angle). If the values are within the scope of your saw, set these values on your miter saw. Some projects are too wide for even a sliding miter saw, so the table saw comes into play. For really large projects involving plywood, for example, you may need to draw the miter angle, M, on the wood and use a skill saw set at B for the bevel angle.

I have a Ryobi BT3000 with a large sliding miter table that is nice for this kind of work. For most table saws it is helpful to make a sliding platform on which you can index the miter angle using my Wall Angle Gauge (Part 4  Special Fixtures and Tools) or your protractor. A simple 1x2 screwed to the sliding platform can be your miter fence. Most, or all, table saws have the blade tilt to the right. If all your cuts are made with the work piece to the left of the blade, it will be necessary to make the second cut after flipping the work piece up-side-down. Be sure to visualize how the edges come together, because the orientation is different if you make the cuts to the right of the blade.

If your design involves shallow boxes (such as my shadow box), you may have to specify the angle g, the width of the board (5.5 inches in my case), the lumber thickness and A, for example. The spreadsheet allows you to enter these numbers to calculate the other dimensions and M and B.

2. Boxes of non-uniform size and shape:

The above calculations are for a box of uniform shape with all sides the same. If you should have a box that is tipped and/or with different size walls, the calculations become a bit more involved. The spreadsheet includes an example of a box as shown below.

The sides of the box are trapezoids; side 2 and 4 have vertical walls, while sides 1 and 3 are tilted at an angle g. The spreadsheet points out that the miter and bevel angles for each piece and each cut depend on the tilt angle g and the appropriate "wall angle" β. The spreadsheet shows how the miter and bevel angles are calculated for each cut. And this brings us to the more universal approach of compound angle cutting:

Always find out what the tilt angle, g, and b are for each cut.

This can often be a challenge. It requires looking at objects in 3D space and then translating this to a two dimensional plane. For this reason I have included this next section. It will also help when we discuss crown molding on vaulted ceilings (Part 3 Crown Molding on Vaulted Ceilings).

3. Planes, Projections and Rotations:

Note: This part is important only if a person wants to know the basis for some of the calculations, or when an unusual geometry is encountered. Crown molding on a vaulted ceiling is one example.

When compound angles need to be calculated, it is important to be able to "see" the cutting surface and reference lines. Drawings and some math are involved. For this we will use 3D models and position them using the three planes X-Y, Y-Z and X-Z.

The diagram below shows how these three planes meet at right angles. When an object is within this framework, it can be projected onto one, or all three, of the surfaces. The sample only shows the projections on the X-Y and Y-Z planes. It is as if a light is held behind the object and casts a shadow on the wall (the light being far away).

Consider an object in 3D space. I have tried to do this by showing three planes defined as the floor or X-Y then two walls Y-Z and in the back X-Z. For our purposes the most important plane is the cutting plane, the plane of the saw blade. For straight cuts that would be the Y-Z plane. But for others it is a function of B and M (the bevel and miter angles).
We have seen that when crown molding makes a turn the important factors that determine B and M are the wall angle and the tilt angle. Sometimes the wall angle b becomes difficult to visualize. It would be much simpler if the part were lying on a surface that is perpendicular to the cutting plane. That is when we can use the concept of object rotation.

Let’s try to visualize this rotation by first looking at a simple line and then a flat board:

Imagine that the cutting plane is Y-Z. Let's look at a line that goes from A to B. Line AB slopes upwards at an angle q and is slanted at an angle r. The line is too flimsy to be cut standing up like that; it needs to lie flat on plane X-Y.
Now let’s rotate the liontoies on the horizontal plane X-Y. Point B moves to C. The length of the new line A-C has to remain the same.  Notice that the angle r is reduced to b/2. The amount it shrinks depends on how large q is. For q equal to zero, the line lies flat on the horizontal plane and the cut is like a miter cut (r and b/2 have the same value). The larger q is, the smaller b/2 becomes. The significance of this will become clear when we look at line A-B as if it were a flat board. For the following we assume that the angles r and q are the same as above.

By rotating the board onto the horizontal plane X-Y (and assuming the vertical plane is the cutting surface), you might be able to visualize that the green board becomes one side of a slanted box. This allows us to calculate B and M (see the section about compound angles and the corresponding Excel spreadsheet). We will not go into the math for this.

Hopefully this explanation is helpful in understanding how the parts meet, how they are cut, and how they make a joint to a surface or mating pieces. If you still have trouble visualizing how this works, try the following: Put your left elbow on a table and point your arm up and to the right as if it is the board. Hold your hand straight and flat. Now pivot your arm to the left and down. Your hand should follow with the thumb up. This shows how it rotates toward the left. You could do the same to the right.

I have one more suggestion about cutting these angles. When in doubt, and if you can, bring the piece to the intersection, hold them in place and, using a pencil, roughly draw the miter and bevel lines as you visualize them. It will help to determine on which side of the saw you will be cutting and in what position. This is especially important with boards that are square or rectangular in cross section.

4. Example of compound angle cuts in construction work:

Going back to our discussion of compound angle cutting, let’s take an example:

Two heavy rafters meet at different angles as shown in the sketch. They are thick, expensive and visible (no trial and error). How do we cut rafter R1 so it makes a good joint with R2?
The plan view (or view from the top) shows angles 20 and 75 degrees for the two rafters. However, they are also tilted at two different angles from the horizontal. Hence, it this a compound angle situation.

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