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.