# Carpenter's trick to portion board into thirds - Approximation or Exact?

In this video (starting at 1:30), a "trick" is presented to mark a board of unpleasant length (call $$l$$) into equal thirds, by angling the tape measure to a more convenient integer length (call $$t$$, defined as $$l$$ rounded up) and dividing that integer into thirds.

My first instinct was this trick must be an approximation, valid only for small angles ($$t \approx l$$). I attempted to derive a formula for the error in the approximation, which I assumed might depend on $$l$$ and $$t$$ (and through them the angle, with $$\cos\theta= \frac{l}{t}$$), and possibly depend on the number of segments the board was being portioned into.

To my surprise, I seem to have shown that the portion width based on the angled trick ($$y$$ in the following diagram) is indeed exactly equal to a third of the board length, regardless of the angle of the tape measure.

My question: Is the "trick" described indeed an exact geometric method, or have I made a mistake in my calculations? Is there a more elegant way to demonstrate the answer than my approach?

Here is my attempt at solving the problem:

Sketch of the geometry

(Please pardon my unorthodox notation for $$sin$$ and $$cos$$.)