Posted by: Dave Richeson | October 19, 2008

## 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.