## Why ‘Divisible By Pi’?

This is an adaptation of an explanatory post I did, taken from its natural environment, slapped around a bit and put here for your viewing pleasure. Enjoy.

We’ll start with the short answer: All of the nicest angles are divisible by pi. If you’ve done any first year university maths subjects, this should make immediate sense to you — the rest of you, likely, are going to require a bit more convincing.

The first question you should be asking yourself is, “How is he measuring angles?” The common understanding of how angles are measured – i.e. in degrees – clearly makes no sense here. A circle has 360 degrees, a number that doesn’t lend itself easily to finding multiples of pi lurking around.

The units that we use to measure the angle – and that work out in the end to be much more convenient and intuitive once you’ve got a good grasp of them, are called ‘radians’ (abbreviated to ‘rads’) and are related to degrees by the simple formula $\frac{\pi}{180} rads = 1 deg$. If this seems like an odd measure to use, multiply both sides of the equation to get $2\pi rads = 360 deg$, and remember the old high school equation for circumference of a circle: $C = 2\pi \times r$. Voila!

The next bit of high school trig to remember are the rules for the sine, cosine and tangent functions commonly abbreviated as “SOH CAH TOA”. For those of you who don’t remember, SOH CAH TOA is a mnemonic for the three equations:

$\sin\theta = \frac{O}{H}$ $\cos\theta = \frac{A}{H}$ $\tan\theta = \frac{O}{A}$

Where ‘A’, ‘O’ and ‘H’ stand for ‘Adjacent’, ‘Opposite’ and ‘Hypotenuse’, as in the diagram on the left. As can be clearly seen in the diagram, adjacent and opposite refer to the position of the sides with reference to the angle theta that we are interested in, and the hypotenuse refers to the long side opposite the right angle in a right angle triangle.

(I know, I know – you remember all this…just bear with me, in case there are people who don’t.)

Now that we’ve got all that down, you might be able to see where I’m going with all this – given that the sine, cosine and tangent functions can be thought of as reporting the ratio of two sides in a right angle triangle (with caveats and tricks applied along the way that aren’t important here), the nicest angles are the ones that report back nice ratios when plugged into a trigonometric function. Aside from anything else, such angles and useful when you have to make approximations – it’s much easier to deal with a simple ratio than a value like $\sin\left(\frac{3}{7}\right)$.

All of this brings me to the final part of the post – this picture:

If you’ve followed me along this far, you can probably guess that this has really just turned into a game of ‘fill in the blanks’. For the triangle on the left, the non-right angles are both $\frac{\pi}{4}$, and for the triangle on the right the bottom angle is $\frac{\pi}{3}$ and the top angle is $\frac{\pi}{6}$. Notice anything in particular?

That’s right. They’re all divisible by $\pi$.

### 2 Responses to “Why ‘Divisible By Pi’?”

1. 3.14159 26535 89793 23846 26433
83279 50288 41971 69399 37510
58209 74944 59230 78164 06286
20899 86280 34825 34211 70679
82148 08651 32823 06647 09384
46095 50582 23172 53594 08128
48111 74502 84102 70193 85211
05559 64462 29489 54930 38196
44288 10975 66593 34461 28475
64823 37867 83165 27120 19091
45648 56692 34603 48610 45432
66482 13393 60726 02491 41273
72458 70066 06315 58817 48815
20920 96282 92540 91715 36436
78925 90360 01133 05305 48820
46652 13841 46951 94151 16094
33057 270….. mmmm,  Pi.  (Homer Simpson)

2. As someone who’s always been interested in pi, I must say: Yes, I want a drink, alcoholic, of course, after the heavy lectures involving quantum mechanics. — A mnemonic for pi. Think about it. Along the same line is this poem: http://jots.org/~ken/Near_a_Raven.html . And, lastly, I do hope you’re aware that there is a vicious assault on pi in the offing: http://vihart.com/blog/pi-is-still-wrong/

Enjoy!