Reply To: Night stand table – No longer square after glue-up

I’m not a math wiz, so I’m curious how you got to your calculations. I’m sure I can find them … somewhere …

Its called the “law of cosines” which gives the included angle C of a triangle with sides a, b, c as

cos(C) = (a^2+b^2-c^2)/(2*a*b)

where Cis the angle included between sides a, b. In our case we have a nearly right triangle with c as the diagonal. What you measure as 1/16″ is the difference between diagonals of a parallelogram[Note 1] that is nearly a rectangle. For small deviations this difference will appear as half that much (1/32″) in each diagonal, one shorter and the other longer than ideal values.

All that put together, and using the fact that deviations are small, and letting a = b for simplicity, gives a simple expression for the angular error in radians as[Note 2]

angular error = 0.7 d/a

(or 0.5 c*d/(a*b) in the more general case of differing a and b).

where d is the measured error (1/16″ in our case). Multiply by 57 (180/pi to be precise) to get the angular error in degrees. For d = 1/16″, a = b = 24″, its about 0.1 degree.

Notes
1. The carcass is a parallelogram provided the pieces were cut accurately to start with.

2. To get that expression use cos(x) = sin (pi/2 - x) and some approximations:
(i) sine of a small angle is the angle itself in radians
(ii) for small deviations d^2/c^2 is much smaller than d/c and can be neglected

3. For narrow and long boxes (a << b or b << a), the angular error is close to 0.5d/a where a is the shorter side. Which could be much larger than d/b where b is the longer side. This may come as a surprise to some.