In celebration of pi

If you happen to be in a bar and you need to quickly calculate $\pi$ , head over to a dartboard. Draw a square that perfectly encloses the dartboard. (The length of its sides should be equal to the dartboard’s diameter.) Start throwing darts randomly at the square.

If you manage to distribute the darts uniformly at random over the square then the fraction of the total darts that land in the dartboard will equal the area of the dartboard divided by the area of the square. If the radius of the dartboard is $r$ , then this fraction will be

$\displaystyle \frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi r^2}{ (2r)^2} = \frac{\pi}{4}$ .

Notice that the ratio is independent of the value of $r$ !

Therefore if you take the number of darts that land in the dartboard (and not in the surrounding portions of the square), divide by the number of darts thrown and then multiply by four … you’ve roughly approximated the value of $\pi$ !

(But remember … it only works if you’re _bad_ at darts.)

While we’re at it, here’s an oldie but goodie. (Not sure of the original source.)