Measuring an angle with a ruler

In the September 2008 issue of the College Mathematics Journal Travis Kowalski presents an neat way to measure an angle using a ruler.  He attributes the discovery to a student of his, Tor Bertin.

Given an acute angle \alpha (the technique can be modified for obtuse angles), measure off a distance s on each ray.  Then measure the distance between these two points, b. He claims that \alpha is approximately \displaystyle\frac{60b}{s} degrees.

He illustrated this technique using s=3.  Some example include:

  • if \alpha=15^\circ, then the approximation is 15.7^\circ
  • if \alpha=45^\circ, then the approximation is 45.9^\circ
  • if \alpha=70^\circ then the approximation is 68.8^\circ
  • obviously, if \alpha=60^\circ, then the approximation is 60^\circ.

As another example, if we take s to be 6 centimeters, then the measurement of b in milimeters is the approximate number of degrees for \alpha.

The derivation of this approximation is elementary.  Using trigonometry, it is easy to see that \displaystyle\sin(\frac{\alpha}{2})=\frac{b}{2s}.  Assuming sine takes angles in radians, but that \alpha is measured in degrees, this becomes \displaystyle\sin(\frac{\pi\alpha}{360})=\frac{b}{2s}. Then the fact that \sin\theta\approx\theta and \pi\approx 3 yields the desired result.

The rest of the article is devoted to looking at whether 60 is the best constant to be used in this approximation formula.