## Using Multiple Moulding Widths In One Frame

revised 6/1/2020

In this article, the first of the “Weird Wood” seriesintro, we show how to build a picture frame using four strips of moulding that aren’t all the same width. Although Figure 1 uses a different width for each piece of moulding, we used three different sizes in our test frames (only because I couldn’t find four different sizes in the same moulding family).

## First The Math

Warning: This discussion includes a little trigonometry.  Do Not Panic! It’s not as bad as it sounds.

### Definition of “Tangent” (skip ahead To Next paragraph if you still remember this):

There are three sides to any right triangle (a triangle with a 90° corner), which I will call the height and the width, which both touch the right (90°) angle and the hypotenuse, which is opposite the right angle and is the triangle’s longest side. You can use the ratio of the lengths of any two of those sides to find the size of the other two angles. Each possible ratio has a name, but we are only interested in one of them today. Probably the most common ratio and the one we will be using is called the tangent. The tangent is defined as the ratio between the height (the length of the side opposite the angle you are interested in) and the width (the length of the shorter of the two sides that create that corner that you are interested in). If you want to know the angle of corner α in the above drawing (Figure 2), for instance, you would calculate its tangent by dividing the height (3 inches in this case) by the width (1¼ inches), which is 2.4 this time. Then you would use your calculator (or phone app – I use RealCalc Plus by Quartic Software (even though it cost 3.50)) to find the angle corresponding to that tangent. On your calculator, the tangent is abbreviated “tan”. If you enter 45 (degrees are assumed) and hit the “tan” button, you will get 1 because for a 45° angle the height is the same as the width, so their ratio is 1. To go the other way (to find the angle), like we are trying to do, we need the inverse of the tangent. Look for the “tan-1” button (it could be the same button, in which case you may need to hit a (yellow) shift or second-function key, and then hit the “tan” button). In this case, once we have the tangent of 2.4, we hit the inverse tangent button(s) to get 67.380135…. (the calculator is obligated to give you 8 or more digits – that doesn’t mean they mean anything. In Figure 2, I rounded that answer to 67.4 degrees and even that third digit should be suspicious.) ## The Process All you have to do is take the ratio between the widths of your two moulding pieces and take the inverse or arc-tangent to get the angle. Here are a few things you need to remember: 1. Which angle – the tangent gives you the angle that was touching the side whose length was used for the denominator (the width, which would be the second number in the division). The simplest way (but certainly not the only way) to get the other non-90° angle is to just subtract the first from 90° (since the two angles are complementary). Also remember that if the tangent was greater than one, the angle will be larger than 45°; if it was supposed to be a smaller angle (less than 45°), then you may have divided the two lengths in the ratio backward. Don’t worry, you just found the complementary angle and all you have to do to get the right answer is subtract what you got from 90. 2. It is up to you to keep track of whether that angle you are cutting should be to the left or the right. Making a drawing of your frame design might help. To be useful, the drawing doesn’t even need to be that good. This should also tell you if you calculated the complement (the other angle in that corner (for the other piece of moulding)). 3. Your saw may be measuring angle backward. My miter saw calls a cut perpendicular across the board 0°, not 90°. If that’s the case, just subtract the angle you calculated from 90. As an exercise, go ahead and check the rest of my calculations in Figure 1. 😁 ### Make The Cuts There is more than one way to make these cuts and more than one set of tools to help you. Which set of tools you should use will depend on such factors as how much of this work you intend to do, your skill set, what your budget is, and what tools you already have on hand. To see the Note click here.To hide the Note click here. Looking through the Framers’ Corner, the forum of the Professional Picture Framers Association, I found recommendations for the following tools for this application: 12-pc Precision Angle Block set (1/4, 1/2, 1 to 5, & 5 to 30 degree) You would only need one of these (if any), not both. (The Amazon.com descriptions are only used here as a reference. Although frequently competitive, Amazon isn’t always the only or the best place to buy something.) Our workshop includes all of the tools listed in www.BeeHappyGraphics.com/about.html#BruceEquip, along with a number of other regular hand & power woodworking tools that Nancy has accumulated over the last several decades. For this project, I used our compound miter saw, but not without complications. To see the Note click here.To hide the Note click here. The precision on this saw looked fine; you should be able to get within ¼° of your target. The first picture (Image A) shows me trying for 22.6° (which would be one of the angles between a 3″ and a 1¼” moulding). After cutting the 3″ piece, I ran into problems trying to cut the complementary angle (67.4°) on the 1¼” piece, as shown in Image B. I am not claiming that mine was the best path to reach our goal. In fact, I would love to see your ideas in the comment section about how to improve my techniques. #### How I Did It 1. Working with one corner at a time, I cut both pieces of moulding square just a tad longer than their overall/outside measurement according to your diagram (you will see why in Step 4). If you don’t already have one, this is also when you would put a perfectly square cut on the alignment block you’ll see in Figure 5 to the left of the moulding. I grabbed a 2″ by 4″, but the wider the better. 2. I set the saw for the smaller of the two complementary angles, rechecking my diagram to confirm whether it should be to the left or right. In the setup shown below, the 3″ moulding would be clamped to the right of the blade. 1. I made the cut. 1. Without adjusting the angle of the saw, I set up the second cut. I positioned my (newly cut) alignment block to the left (opposite the side we placed the moulding for the cut (in Step 2)) so that I could also place the 2″ moulding to the left of the blade and perpendicular (at a right (90°) angle) to the miter saw fence. After clamping down the alignment block, I added a support block to the right of the moulding to keep it in place. I could still move the moulding in or out to position the cut. As you can see, I needed to precut this piece of moulding to keep it from extending too far into the aisle and getting in my way. Another reason is explained in Step 6. 2. I made the cut. To see the Note click here.To hide the Note click here. For those who noticed that the color of the moulding in Figure 4 was different than in Figure 5, I had to make two different frames while doing research for this article 1) to confirm and refine my techniques and 2) because I didn’t get enough pictures the first time. 1. Always check your work. Since you precut each piece of moulding a little longer than necessary, consider this first cut on each piece a test cut. See if the two pieces match up as expected. If, when you put the two pieces of moulding together, the miter edge on one piece is longer than the other, that is the angle that should have been larger. The angle on the other piece of moulding should have been smaller (by the same amount). Figure 7 shows the second setup from the right side. If you look close, you might notice that I didn’t cut enough to make a sharp corner and needed to recut. 1. Moving to the next corner, I precut another piece of moulding and repeated Steps 2 through 6. Now that both angles are correct for the second piece of moulding, you can recut its last miter if necessary to get the length right. 2. I repeated Step 7. When both angles are correct on the third piece of moulding, recut the last miter on that piece as necessary to get the length right. 3. I repeated Step 7 one last time. When both angles are correct on the fourth piece of moulding, I used the second piece of moulding to mark the length of the fourth piece by matching the inside edges, as shown in Figure 8. Similarly, I used the third piece of moulding to mark the length of the first piece. ### Finishing As with my normal (45° miter) frames, I would next need to make sure the inner lengths on opposite pieces of moulding matched, and the outer lengths as well. Figure 9 shows a way to check to see if the outside and inside corners of the opposite sides match using two carpenter squares (or equivalent). Some of the tools we normally use next to finish putting the frame together, namely our Logan Precision Sander and Logan Pro Joiner, are worthless for this application. After gluing (and clamping the pieces together until dry) we had to pound the V-nails in by hand (interestingly, the simpler Logan Studio Joiner can be adapted). ## The Back Side For completeness, the left figure below shows what the backside of the lower left corner would look like. The gray section represents the rabbet, the equal-width (¼”) cut-out that holds the glass, mats, image, and backing of the picture inside the frame. Some of you might be surprised to see that there is a triangular notch in this rabbet in the corner along the miter cut. This notch has no effect on the functionality of the rabbet. To solve this “problem”, however, you could make a compound cut 45° in from the inner edge to the edge of the rabbet and 79.7° in from the outer edge to the same point, as shown in the right figure below (as an exercise, you can check my math on these angles also). But there is really no need to make these cuts. If the gray were to represent an equal-width feature on the front of the moulding, it might be worthwhile to take the extra trouble. Otherwise, don’t even think about it. ## The End Congratulations, you now have a fancy new picture frame. Of course, you still need to find a picture, cut mat(s) and backing, mount picture to same, cut glass, assemble the pieces without showing any annoying little specks, and apply a dust cover and hanging hardware, but all of that is beyond the scope of this article. Good luck! As mentioned, this article is just the beginning of a series about “Weird Wood” that I announced months ago. Up next, we will look at handling moulding that is not of uniform width. You won’t find this moulding in any store; it is only an exercise to prepare you for our final project. But if it stimulates your creativity, that’s not always a bad thing. Stay tuned, and thanks for reading! Your comments are welcome and appreciated. ## Comments on Mat(h) Solution I posted our first Simple Mat(h) Problem on April 27, 2017, and Jim Farrington submitted a solution a couple of weeks later. Here are a few more comments on the problem that I published (but in the wrong place). Although the mat cutter has no kerf, at the start of the cut the blade does swing down and could cut into the side of your finished piece around that square in the middle. For backing boards, this is not a problem, and because at least 1/8” will be hidden by the moulding, it is probably not a problem when cutting mats either. There is enough extra space that if you wanted to play it safe you could put a 1/2” between the four pieces as shown below. ## Could There Be More Than Four Pieces? To be blunt, NO. There just aren’t enough scraps to possibly make another 16″ by 20″ piece. Five such pieces would have an area of 5 x 16″ x 20″ = 1,600 square inches. We started with a piece that was 39″ x 37″, or 1,443 square inches. It just can’t be done. ## Is There Another Way To Get The Correct Answer? We’ve all managed to get a broom into a shorter closet by sliding it in at an angle. One question I had was “Would it be possible to squeeze the originally planned 40″ by 32″ rectangle needed for the four pieces into the 39″ high space by rotating slightly?” The short answer is NO. Because this is so much wider than a broom, as you rotate, the required height actually gets larger at first and doesn’t drop back below 39″ until you’ve rotated over 79°. By then the necessary width would be over 45″ (well over the 37″ available). To see the math, see the note below (Warning: this “solution” requires knowledge of trigonometry). Of course, this doesn’t guarantee that there is no other solution. If you find one, let me know. To see the Note click here.To hide the Note click here. To help with the math, I made a drawing. The target 40″ by 32″ rectangular piece of foam board is shown by the dark blue rectangle. The red rectangle is the smallest “box” that can be placed around it. The length (L) of the outer box can be described as $L = l \cos \theta + w \sin \theta$. This reminds me of a trigonometric identity for the sine or cosine of the sum or difference of two angles: $\sin (x \pm y) = \sin x \cos y \pm \cos x \sin y,$ and $\cos (x \pm y) = \cos x \cos y \mp \sin x \sin y.$ Let d be the length of the diagonal of the inside rectangle. It can be found as $d = \sqrt{l^2 + w^2}$. Furthermore, $\frac{l}{d} = \sin \alpha$ and $\frac{w}{d} = \cos \alpha$, where α is the angle that the diagonal makes with the base of the rectangle. If we divide both sides of our first equation for L by d, we get: $\begin{array}{r c l} \frac{L}{d} & = & \frac{l}{d} \cos \theta + \frac{w}{d} \sin \theta \\ & = & \sin \alpha \cos \theta + \cos \alpha \sin \theta \\ & = & \sin (\alpha + \theta) \\ L & = & d \sin (\alpha + \theta) \end{array}$ By the same token, the width (W) of the outer box can be described as $W = l \sin \theta + w \cos \theta$. Following a similar path as above, we can show that $W = d \cos (\alpha - \theta)$. Now you can plug 39″ in for L, 51.225″ (which is square root of 402 + 322) for d, and 51.34° for α (by taking the arcsine (which may be shown as sin-1 on your calculator) of 40″/51.225″) into the last equation for L, and use your calculator to find that α + θ = 49.583°. Unfortunately, that’s less than α and doesn’t meet our requirements. But around a circle, there are two angles with the same sine. Your calculator will find the one that is less than 90° (we’ll call it 90° – φ). The other would be 90° + φ, which makes θ = 79.192°. Plugging that, along with d and θ, into the last equation for W gives the result I mentioned above. ### A Shortcut? That last equation for L could also have been found more directly by noticing the orange right triangle formed by the dashed orange line along the inside blue rectangle’s diagonal and through the lower right corner of the inside rectangle (as the hypotenuse), and the vertical dashed red line of length l (as the side opposite the angle), and the segment of the bottom edge of the outer rectangle that’s between those two other sides (as the adjacent side). The light blue arrows around that lower right corner show the angle of this right triangle is α + θ. Then by noticing the yellow right triangle formed by the other orange dashed diagonal line as hypotenuse and the other red dashed line of length w as the opposite side again we could have found that $W = d \sin(\theta + (90 - \alpha))$. Then, since $\sin x = \cos(90 - x)$, $\begin{array}{rcl} W & = & d \cos(90 - (\theta + (90 - \alpha))) \\ & = & d \cos(\alpha - \theta) \end{array}$. Now, aren’t you glad you asked? ## Another Method For Adjusting A Logan Sander To download a printable version of this article (AdjustingSander.pdf), click here. We have a Logan Precision Sander Elite Model F200-2 disk sander for improving saw-cut miters for your picture frames to a “perfect 45°” after cutting the moulding to size on our miter saw. To maintain such perfection requires due diligence and occasional adjustment. ### How Do You Know When It’s Time To Adjust Your Sander? 1. You may notice that when you put your frames together, there is a small gap between the pieces of moulding either on the inside of all four corners or the outside of all four corners. If some corners have gaps on the inside and some have a gap on the outside, you have other problems. In the figures used in this article, the symptoms have been exaggerated for illustration purposes. If the condition of your sander gets this bad without you noticing, you may want to consider another profession or hobby. 1. When you are comparing the lengths of opposite pieces, and you have them side by side with the miters face up and their back sides touching, you may notice by running your finger over the miter that they are the same length on one end of the miter but not the other, or that one piece of moulding is higher at one end of the miter and the other piece is higher at the other end. In either of these cases, it’s time to adjust your sander. ### But What About The Miter Saw? It may be true that the miter saw also needs adjustment, but that would have minimal impact on your frames because even if the angle of the cut was wrong, the sander should correct that problem. Of course, it would take more sanding to correct, which besides taking more time and effort could, in the worst case, result in your frame being too small, so it should periodically be checked and corrected according to the manufacturer’s instructions (I currently have no improvements or suggestions for that process). An indication that the miter saw needed adjustment would be if as you are sanding the miter, sawdust builds up on top of one side of the moulding faster than it accumulates on the other. If it takes too many revolutions of the sander to perfect the edge, that could also be a clue, or it could be time to change the sandpaper. ### How To Adjust The Sander On the last page of the 4-page manual (available at www.logangraphic.com) are simple instructions for that adjustment that should work well if you are willing to follow Step 1 and remove the sandpaper. To see the Note click here.To hide the Note click here. The full instructions are as follows: Adjustment 45° 1. Remove sand paper 2. Place the 45˚ square flat against the wheel and up against bar (Fig. 7). Look for gaps against the bar. 3. Adjust the bar using adjustment wrench until gap disappears. When I don’t remove the sandpaper disk the technique doesn’t work as well, so I’ve come up with an alternate set of instructions: 1. Put miter cuts on both ends of two long scrap pieces of your widest moulding. 2. When you sand a piece of moulding, each end will use a different side of the sander. Call one side of the sander “A” and the other “B”. As you sand the two pieces of moulding, mark the back of each end of each piece with the side of the sander used (A or B). 3. Find a good right angle, either in a reliable carpenter’s square or using other methods. 4. Flip one of the pieces of scrap moulding upside down so you can join Corner “A” on both pieces to make a 90˚ (right) angle. Flipping is very important*. 5. Put one piece of moulding along one edge of the reference angle (carpenter’s square) and slide the reference toward the second moulding until it just touches at one end or the other (if it touches at both ends, you are finished with Side A – skip ahead to Step 7). Measure the error gap (I like millimeters only because they are so small) at the end of the moulding that’s away from the reference line. Then measure the length of that piece of scrap moulding (using the same units of measurement). 6. The adjustment screw on my sander had 32 threads per inch, and it was 109 millimeters from the pivot point. Based on that, your multiplier will be 25,000. To see the Note click here.To hide the Note click here. I’ll do the math just in case your sander has different measurements so that you can substitute the real numbers in for mine at the proper places. One complete turn of the adjustment screw is 360° or 1/32” and there are 25.4 millimeters per inch, so the constant multiplier would be 109 mm * 360° per turn of screw * 32 threads per inch ÷ 25.4 mm per inch ÷ 2 errors = 24,718.11. We’ll say 25,000. Divide the error you measured in the last step by the length of the moulding that you measured and multiply by 25,000. Your result will be the number of degrees you need to turn the adjustment screw. If the error gap was at the corner, then the angle is too large and you have to turn the screw counterclockwise to back it out. Conversely, if the error gap was at the end of the moulding, then the angle is too small and you have to turn the adjustment screw clockwise to push it out more. After making the adjustment, you may want to retest by repeating the process by starting at Step 2 and resanding the same two corners just used. Before sanding, I recommend drawing a line all the way across the end of the moulding with a pen or marker and then sanding until the line completely disappears. 7. Repeat this whole process (starting at Step 4) for the other two miters, labeled “B”. Remember, for these two you will be playing with the adjustment screw on the other side of the sander. That’s all there is to it. Congratulations. To see the Note click here.To hide the Note click here. #### Math Warning: a quick note about trigonometry (OPTIONAL)! This process was concerned with angles, not distances, but since angles are harder to measure with any precision we had to convert. When you take the ratio of the two perpendicular sides (the sides that are 90° apart) of a right triangle containing the angle you are interested in, that’s called the tangent of that angle, and you can have a good calculator app on your phone find it for you (for my Droid, I found RealCalc Plus by Quartic Software at the Play Store and was happy to pay3.50. There are plenty of other options, though).

The error angle you measured (indirectly) was actually twice as large as the real error. One problem is that the tangent curve is not generally a straight line, which means that the tangent of twice some angle is not the same as twice the tangent of that angle. That’s why the normal procedure would be to convert to angles, do the adding, subtracting, or multiplying, and then convert back to distances we can measure again. We were able to use the small angle exception, however. It turns out that for angles less than say 10°, the tangent curve IS pretty straight and the error introduced by taking our shortcut isn’t worth worrying about. That’s what we did, and that’s why the problem was so easy. That’s your math lesson of the day week month. Let’s get back to work.

## An Optical Illusion

When we do the gallery wrap on our canvas prints, I usually take just a tad along the edge of the image and digitally stretch it to several times its original width to wrap around the edge of the frame. When we do that stretching, we set up an optical illusion. When the image is hanging on the wall and you move to the side of the image, there will be some angle at which the stretched edge image will seem to be just a continuation of the image on the face of the frame. When I’ve discussed this with people in our booth at art festivals, I’ve left the calculation of that angle as an exercise for the listener.  Now I’ll describe the solution of that exercise in an article “Finding the Angle of the Illusion”.  This article is now on our website, but can’t yet be reached through the menu system. (I’ve recruited a friend to help me update the website, which is long overdue, so please bear with us).

Caution: this solution does involve basic trigonometry. Who would have thought we would ever use that stuff?