Making Cove Mats

I recently listened to a member of the Professional Picture Framers Association (PPFA) talk about how a computerized mat cutter (CMC) can make it easier to cut cove mats without getting into the math (since the consensus was that it would involve trigonometry). I don't own a CMC, and can't justify buying one, but I took the cove mat problem as a challenge. Turns out you don't need anything more complicated than finding the hypotenuse of a right triangle. In this article I explain how to make those mats even without a CMC.

What's a "Cove Mat"?

Not to be confused with "cove rectangles"^Defined, cove mats, as used in this article, are essentially shadowboxes^Defined with sloped sides. A few photos can be found at The Picture Framers Grumble (forum)^Link.

Note: There is some controversy about the use of the term "cove mat" among professional framers. Although many picture framers call those like the one discussed in this article a cove mat, purists insist this is only a "tray mat", since true cove mats have curved sides. I have never seen a 'true' cove mat, and could not even find a photograph. Those would be well beyond the scope of this article. Nonetheless, in future articles, I will conform to the stricter convention and use the term "tray mat."

How To Make It

To use these instructions, all you need is the depth of the box, as shown in the below side view, and what I've called 'the spread', or how much wider the box needs to be at the top than at the base along that edge. As hinted in the Figure 1 side view, you can have a different spread along each of the four edges, although that would be rare. The red numbers are for the example we will be using in this article, and the brown numbers were derived from them as described. Both the Depth and the Spreads are inside measurements. To get outside measurements for whatever reason, just add the mat thickness to the depth and twice that much for each of the two side dimensions.

Our example has a 5" by 7" base. The top and right sides have a 1" spread (S_T and S_R) so we can show the conventional corner treatment. To demonstrate that the process is versatile, the left edge has a 2" spread (S_L), and the bottom spread (S_B) is 3". The finished overall length will be 1" + 5" + 2" = 8" (plus 2 times mat thickness), and the height would be 11", so let's get started.

1. Decide Depth (D) and Spreads (S).
2. Calculate the Wall Widths (W). This is the most complicated step, and as you've already guessed, it involves the Pythagorean Theorem. (Or does it? See Analog Hypotenuse.) In our example, where the depth (D) is 2" and the spread on the right side of Figure 1 (S_L) is also 2", the wall width,

$$W_L=\sqrt{D^2+S_L^2}=\sqrt{4+4}$$

≈ 2.828 or 2¹³/₁₆.

3. Cut a Piece of Matboard.  For what it's worth, David Lantrip, MCPF, GCF, the speaker at that PPFA^Website CMC session, recommended fabric mats for this application.

The mat's overall length will be W_R + L + W_L. For our example, that would be about 2¹/₄" + 5" + 2¹³/₁₆" or 10¹/₁₆" (compared to the finished cove overall length of 8", as discussed above). If you were wondering why W_R was on the left side of Figure 1, remember you will be measuring, marking, and cutting on the back side of the mat, as usual, so the image has been flipped.  That means that the spreads appear to increase from 1" to 2" to 3" as you go around the edge of our example mat clockwise instead of counter-clockwise.  If all of your spreads are the same, this isn't an issue.

Similarly, the overall height will be W_T + L + W_B. In our case, that's about 2¹/₄" + 7" + 3¹⁹/₃₂" or 12²⁷/₃₂" (12⁷/₈" should work).

4. Draw Base Rectangle. Measure and mark the appropriate W_? in from each edge (on the back side of the mat, as discussed). Use a straight edge or ruler to draw the edges. The lines need not extend beyond the corners of the rectangle.
5. Draw Spread Construction Lines. Measure and mark the appropriate spread (S_?) out from the base rectangle along each edge and draw the construction line. Extend the construction lines out to the edges of the mat. In our illustrations, the construction lines are the dashed lines. The yellow portion represents the final position of the upper edge of the cove/tray wall and thus shows the finished overall length and width of the cove mat.
6. Draw Corner Edge Cut Lines. From each corner, start each of the edge cut lines where the nearest spread construction line hits the edge of the mat and draw the straight line to the nearest corner of the base rectangle. Repeat along both edges of that corner. If done correctly, both of these diagonal cut lines at each corner will be the same length. Otherwise, the corners won't match when you fold them up in Step 9. If they aren't the same length, you most likely miscalculated the wall widths in Step 2.

Finding Corner Distance (C) Computerized mat cutters (CMC's) may just calculate the distance from the corner without drawing construction lines. As one can see in Figure 1 (where, again, D is 2", S_R is 1", and thus W_R is about 2¼), that distance, represented by C_Tr in the upper left corner of the Back of Mat view in Figure 1, would just be W_R - S_R, which is

$$\sqrt{D^2+S_R^2}-S_R\approx 1\frac{1}{4}\text{.}$$

7. Cut Corner Edge Lines. Instead of cutting both edges all the way through, one could make a half-cut (the technical term for a cut that doesn't go all the way through the mat) on one of the cuts in each corner so the still-attached material could be used as a backing to be glued to the back of other wall 'flap' and then trimmed during assembly. You may want to make a practice cut on a scrap mat to be sure the cut doesn't go too deep (or too shallow). That's just a thought. In Figure 1, these sample half-cut candidates are represented by black/green dashed lines.
8. Make Half-cuts Along the Edges of Base. This is just to make folding easier and could be considered optional. These half-cuts are represented by the solid green lines in Figure 1.
9. Fold and Assemble the Cove Mat. Use either glue or tape, to taste. And that's it; you're done.  Congratulations!

Other Thoughts

Finding Theta (θ)

If you need to make support for your cove mat (as shown in this post in The Grumble), you would need to know the θ's, as pictorically defined in the Side View at the top of Figure 1. That does take trigonometry. Fortunately, even the calculator in Windows 11 has trigonometric functions now, or you can get a free phone app with that feature.

tan(θ_?) = S_?/D.      Therefore, θ_L = tan^-1(²/₂) = 45°.

Note: For those unfamiliar with the notation, the "-1" superscript after "tan" (or "cos", etc.) means "inverse" (a.k.a. arctangent) or "angle whose tangent (or cosine, etc.) is . . .". Your new calculator has a button to bring those up.

Finding Alpha (α)

If you were cutting this cove out of a door skin, thin plywood, acrylic, or Sintra^About, you would also need to know these angles (shown in the upper right corner of the Back-of-Mat view in Figure 1 above). It can be shown that

$$\cos \left({\alpha }_{\mathrm{Tl}}\right)=\frac{\mathrm{-}{S}_L}{\sqrt{S_T^2+S_L^2+D^2}}=\frac{-2}{\sqrt{1^2+2^2+2^2}}=\frac{-2}{3}$$

so α_Tl = 131.8°. Similarly,

$$\cos \left({\alpha }_{\mathrm{Lt}}\right)=\frac{\mathrm{-}{S}_T}{\sqrt{S_T^2+S_L^2+D^2}}$$

In the most common case where all of your spreads are the same, this formula reduces to

$$\cos \left(\alpha \right)=\frac{\mathrm{-}S}{\sqrt{2S^2+D^2}}\text{ or }\tan \left(\alpha \right)=\frac{\mathrm{-}\sqrt{S^2+D^2}}{S}\text{.}$$

(Just a reminder, you only need to use one of those equations.)

Note: Since α is guaranteed to be obtuse (more tan 90°, but less than 180°), its cosine and its tangent will both be negative.  If you use the equation with the tangent, your calculator will most likely give a negative angle for the arctangent.  Just add 180°.

And if the material you use has any thickness, you will also need to know what angle to tilt your table saw (or miter saw, or radial arm saw) blade. That is also beyond the scope of this article (for now).

What About a Non-Rectangular Base

If the angles on your corners are not 90°, the process becomes at least one step more complicated. Trigonometry is still not absolutely necessary, but a downloadable spreadsheet is available (and can even be used with the simple cove mats we've discussed here).  Those details are explained in Making Tray Mats With Non-rectangular Bases.

Conclusion

So, we've shown that making a simple cove mat requires minimal math. Enjoy!

You can contact us if you have any questions, or hit the Comment Button below if you have any suggestions for improvement.