# Calculating length of sides of angled box

I'm building a cabinet with 19mm plywood. The left vertical side is angled at 60 degrees. I am trying to find out the correct length I need to cut both the vertical and top boards, marked A and B in the picture. Using a 3D software tool, I have calculated that the inner sides 136(A) and 182(B).

Could someone please advice on what the correct length is for outer sides A and B? Will I set the mitre cut at 60 for all angles? All measurements are in mm.

• sin (60) = 138 / A ... A = 138 / sin (60) Just make sure your calculator is in degrees, not radians! – SaSSafraS1232 Feb 21 '19 at 0:41
• Graphus, thank you for input. This is very helpful. Unfortunately, I'm unable to tick this as the correct answer because of my low rep. – Dave Feb 21 '19 at 10:18

You don't need to calculate it, you don't even need to measure it* — you can mark directly from the work.

This has long been the standard way this sort of thing is done in carpentry and woodworking, from before the days of widespread numeracy, the availability of calculators, and now 3D software. Make the three-sided box (doesn't have to be glued together yet), then simply 'offer up' piece A to the open end and mark it for length. This is basically the traditional way drawer fronts were sized to their openings.

This will be accurate enough, as it always has been. If nervous about nailing the perfect size with the first cut you can saw on the waste side of the line and then trim to a perfect fit. In the past this trimming would invariably have been done by shooting with a plane, now you might sneak up on a perfect fit on the table saw or using a belt sander.

For some wiggle room if need be piece B can be cut slightly overlength with either the mitred or square end left to overhang, whichever you prefer. Then take it apart and trim to final length by whatever method you like, or flush in place by planing or with a belt sander.

*Although this is very easily done by drawing full size or to scale and measuring from the drawing. You can draw on the 19mm plywood you're using if you have no paper to hand!

Graphus has the right answer. It is best to mark the work piece from the other three sides of the box to accurately find the correct length. That way if any work up to this point is slightly off — if the three existing sides were cut ever so slightly shorter or longer than intended — it will not matter.

However, it can still be useful to know what the length of the side should be in theory. You can use that to plan rough cuts. Graphus correctly notes that you can get this by printing out to full scale and measuring it. I thought it might be useful to also include the way to do this with trigonometry.

I've taken the original drawing and added a few extra parts that will be helpful. First, the red line, labeled C, and the green line, labeled D, combine with A to form a triangle. The yellow box indicates the angle is 90°, which makes this a right triangle. If we have the length of one side of a right triangle and know one of the other angles besides the one that is 90°, we can use trigonometry to find the length of any other side.

We do know the length of C, because it is the same as the length of the far side, which is 138 (or is it!? see note below). We also know the value of angle AD, which is 60°. Since C is on the opposite side of the triangle from angle AD, we can use the sine function. For any given angle θ,

Sin(θ) = [Length of leg opposite to θ] ÷ [Length of hypotenuse]

The hypotenuse is the longest side of a right triangle. It's the side opposite the right angle. In our case, side A is the hypotenuse.

Plugging in our numbers:

Sin(60°) = 138 ÷ A
0.8660 = 138 ÷ A
0.8660 * A = 138
A = 138 ÷ 0.8660
A = 159.35

A couple of things to note about the math, for completeness sake. First, if you know the value of two angles of a triangle, you can find the third, because in all triangles, the three angles add up to 180°. Second, if you're using a calculator, beware your angle units, as SaSSafraS1232 noted. If you're working in degrees (which we woodworkers pretty much always are!) make sure the calculator is set the same way. The sine function will give different values for radians. (It's kind of like Fahrenheit vs. Celsius — the same temperature can be represented with two different values depending on your choice of units. Angles are the same way, but for degrees and radians.) Finally, if you happen to know different angles or the value of different sides than the ones listed here, you can use a different trigonometry equation. Sine, cosine, tangent, secant, cosecant, etc. There are a lot of them. Look up the definitions and find the one you need.

Or just use an online right triangle calculator such as this one. Just be sure your angles are in degrees, and that you have the sides in the right place.

Final note. In math, lines are one-dimensional. They have no width or depth. In woodworking, our boards and other materials are three-dimensional! Using this diagram, it might be easy to conclude that the length of B is 273 - D. But that would be wrong! The length of B is also must include the width of side A as well as the width of the side labeled "138". No widths are marked on this diagram. It's easy to start working with ideal shapes such as points and lines, and forget to account for the extra width of the material.

The best way to do that, in my opinion, is to not rely on math for the dimensions of final cuts. The math is to help you decide how to break down boards or sheet goods into rough cuts with plenty of margin for a final cut — say, maybe 1" or 2 cm or so on each side. So for example, if I've calculated A to be 159.35 mm, I might do the rough cut to 160 mm + 40 mm = 200 mm, giving me about 2 cm on each side for a final cut. Or if I have started with one edge I trust is straight, I might do the rough cut at 180 mm, leaving myself 2 cm for the final cut. But regardless of how you do it or how much margin you want to give yourself, always make final cuts using the method Graphus described in his answer. Mark, don't measure.

Note about the length of C

Is the length of C really 138? Well, that depends. Is 138 for the inside dimension? If so, then yes, C is 138. But if 138 is instead the length of the board, then C is 138 - [width of top] - [width of bottom]. See how the 3D nature of our material can trip us up? Be careful!

But also, this is another good reason to give ourselves plenty of margin in the rough cut. If you're working with 19 mm plywood and you gave yourself 20 mm per side, forgetting to account for the widths would lose you some of your margin, but you'd still be okay when it comes time to make the final cut.