This Instructable will explain how to make a simple triangular setup tool to help position auxiliary fences on a table saw in order to cut cove profiles, i.e. cove molding or cove panels.

Following this introduction you will see the steps/sections for...

- Constructing the setup tool

- Setup for cove molding

- Setup for cove panel

- Cutting the cove

- Since this was published under the 2019 Instructables Math Contest there is a step/section for the Math

- and finally an "Epilogue" with some miscellaneous information.

Thank you for reading.

Hope you enjoy this Instructable.

Comments and feedback are appreciated.

Step 1: Constructing the Setup Tool

Figure 1 shows the key dimensions for cove molding and Figure 2 shows the corresponding dimensions for a cove panel. That is …

d = depth of cove

w = width of cove for panel, or ½ the full width of cove for molding

In addition to the width and depth of the cove we need to know the diameter of the saw blade and the thickness (kerf) of the saw blade being used.

D = diameter of saw blade

k = saw blade kerf

With these dimensions you can now construct a right angled triangular setup tool to aid in positioning the auxiliary table saw fences. Start with a rectangle with length and width dimensions calculated using the equations below.

width = 2bF + kw

length = 2bw - kF

where F=sqrt( b^2 + (k/2)^2 - w^2 )

and b^2 = Dd - d^2, b = sqrt(b^2)

Then cut diagonal across opposite corners to produce the triangular setup tool.

The derivation of the equations for the width and length dimensions is presented later in the Instructable.

I have included a picture of these equations showing standard mathematical symbols.

Example:

cove depth, d=0.5

cove width, w=1.5

saw blade diameter, D=10

saw blade thickness (kerf), k=1/8=0.125

star by calculating b^2 = (10) (0.5) - 0.5^2 = 4.75, b = sqrt(4.75) = 2.179449

then F = sqrt( 4.75 + (0.125/2)^2 - 1.5^2 ) = 1.582373

and length = 2(2.179449)(1.5) - (0.125)(1.582373) = 6.340550

and width = 2(2.179449)(1.582373) + (0.125)(1.5) = 7.084902

You could cut a triangle with these dimensions but it is probably more convenient to set one of the dimensions to a nice specific value and adjust the other accordingly. For example, set length to 12 and adjust width by ...

width = 12 ( 2bF + kw )/(2bw - kF)

Example:

length=12

width = 12 ( 2bF + kw )/(2bw - kF) = 12 x 7.084902 / 6.340550 =13.408745 or approx 13 13/32

Step 2: Setup for Cove Molding

Set position of first fence

Cut a gauge stick equal, in thickness, to dimension g in Figure 1.
Set height of saw blade equal to the depth of the required cove (d).
Lay the setup tool’s long side (aka hypotenuse) along the saw blade.
Lay gauge stick along the "fence" edge of the setup tool.
Lay the auxiliary fence along the gauge stick.
Holding the setup tool, gauge stick, and fence firmly together slide all along the table (keeping the setup tool’s long side against the saw blade) until the gauge stick makes contact with the saw blade.
Secure the fence to the table top.

Set position of second fence

Remove the gauge stick and setup tool.
Lay a piece of wood (with the same width as the stock you are using for the molding) along the first fence.
Lay the second fence along the other side of the piece of wood and secure it to the table.

Step 3: Setup for Cove Panel

NOTE: Since the auxiliary fence will be position at the mid point of the saw blade (bullet 5 below), the fence must have an opening to allow for the saw blade.

Set height of saw blade equal to the depth of the required cove (d).
Locate the center of the saw blade.
Lay the setup tool’s long side (aka hypotenuse) along the saw blade.
Lay the auxiliary fence along the edge of the setup tool.
Holding the setup tool and fence firmly together slide all along the table (keeping the setup tool’s long side against the saw blade) until the fence is aligned with the center of the saw blade.
Secure fence to table top.

Step 4: Cutting the Cove

After the fence(s) are secured, lower the blade height to 1/16” above the table. Recommend smaller bites for hard wood or wide or deep coves.
Feed the stock over the blade. It is recommended to use feather boards to hold the stock firmly onto the table.
Cut all moldings, or all sides of panels, with the saw blade set to the current height.
Raise the blade 1/16” (or less) and repeat bullets 2 and 3 until all molding, or all sides of the panel, are to the desired depth.

For my prototype I used a scrap piece of hard board for the setup tool, scrap 2x4 and 2x2 for the auxiliary fences, and scrap 2x4 for the project piece. The dimensions of the prototype setup tool were based on the example presented earlier, except I set length = 6 instead of 12, and adjusted width accordingly.

I leave it to the reader to finish the cove by cutting the beveled edges.

NOTE: When you are getting close to the final cove depth I suggest raising the blade less than 1/16", and when at the actual final depth pass the wood over the saw blade a few times. This will create a cleaner cove.

NOTE: FOR SAFETY …

USE GOGGLES
USE DUST MASK
USE PUSH STICK(S)
KEEP HANDS AWAY FROM SAW BLADE AREA
DO NOT WEAR LOOSE CLOTHING
UNPLUG SAW WHEN DOING SETUP AND WHEN PROJECT IS DONE
WORK IN A WELL LIT AND VENTED AREA

Step 5: The Math

Now the math, this is the fun part :)

This section details how the equations for the setup tool’s length and width are determined.

Figure 9 shows the geometry of the table saw and setup tool as seen from above. The required vertices are labeled A, B, C, D, and E.

Conventions used in the solution are

letter triples (e.g. EAB, (EAB)) refer to angles in Figure 9. e.g. EAB and (EAB) refers to angle EAB
letter couples (e.g. CB, (CB)) refer to line segments in Figure 9. e.g. CB and (CB) refers to line segment CB.

The next two pictures show the first few steps in the solution with standard mathematical symbols.

Figure 10 shows a schematic representation of a table saw seen from the side. The height of the saw blade (above the table) is equal to the required cove depth (d). b is half the cord (line where the table top intersects the saw blade).

The next two pictures show more steps in the solution with standard mathematical symbols.

I have also attached a PDF document with the details of the math.

Mathematics of Cove Molding and Cove Panels 2s.pdf
Download

Step 6: Epilogue

For future reference write the information related to the cove on the face of the setup tool. That is…

cove depth = ___
cove width = ___
saw blade diameter = ___
saw blade kerf = ___

Also, label the edges of the setup tool so it is easy to identify how it should be aligned for use. e.g.

“saw blade” for the side that goes against the saw blade
“fence” for the side that goes against the fence
“cove” for the side that represents the width of the cove

You can label the gauge stick too.

Due to the setup tool’s triangular shape you can flip it (while still keeping the same edge against the saw blade) and use it to create another cove of width w' = w ( 2bk + kF ) / ( 2bF - kw ), picture of equation showing standard mathematical symbols is attached.

Using the values from the previous example above this would give a cove width of w' = 1.67578 or approx 1 43/64

Label the opposite side of the setup tool for this complementary cove width.

Simplified Equations...

If the cove width (w) is large compared to the saw blade kerf (k), or if by preliminary estimates you know that the fence(s) will be fairly close to perpendicular to the saw blade, or depending on how accurate your project needs to be, then the saw blade kerf can be ignored i.e. k=0.

In this case the equations simplify to…

length = w

width = sqrt(Dd - d^2 - w^2)

Using the example from earlier (i.e. w=1.5, d=0.5, D=10)

length = 1.5

width = sqrt(10(0.5) - 0.5^2 - 1.5^2) = 1.581138

As before lets set length to 12 and adjust width accordingly...

length = 12

width = (12 / 1.5 ) x 1.581138 = 12.649104

This compares to width = 13.408745 when the saw kerf is not ignored, which is 1.060065 times longer, or ~6% longer. In this case I would not ignore the saw kerf. It is up to the reader to decide when to ignore the kerf and use the simpler equations.

Limitations …
Since the cove is cut from the diagonal projection of a circular saw blade the cove will actually be elliptical in shape. As the setup tool’s width becomes small with respect to the tool's length the cove will become less elliptical (and more circular).

As the setup tool’s width becomes large with respect to the tool's length the fences (and project wood) will be running more parallel to the saw blade. The cove will be highly elliptical and in these cases the kerf of the blade could have an impact on the shape of the cove (IE. it could slightly rectangular).

Other projects …
The following are a few examples (pictures attached) of other projects that could be constructed using this method. The full details for these projects are left for future Instructables.

cove frame
cove track, e.g. marble run tracks
grooved pillars
etc. - be creative, be safe.

More ...

I have attached a PDF document containing tables for three common saw blade diameters (7.25, 8.25, and 10) and for various cove depths and cove widths. The tables also include a rows that specifies the maximum cove width possible for the specified cove depth.