# A geometrical approximation for π

If you were paying very close attention last week, you’ll have noticed my attempt to come up with an estimate of π, geometrically, as part of The Aperiodical’s π Day challenge (even if it’s not really π Day):

Viewed on its own, that’s probably a bit mysterious, so I thought I’d write a little article to explain what was going on, and explore some of the maths behind it.

### The approximation

The first thing I did, after watching the video, was think about (read: Google) what other strategies one might use to approximate π. I thought about measuring a cylinder (too much work); I thought about something to do with the Catalan numbers (would involve research); and I finally settled on a geometric method that relies on a cool almost-identity:

$x \approx \frac{3 \sin(x)}{2 + \cos(x)}$

There are two immediate questions:

• Why does that work?
and
• How does it help?

Why it works is fairly simple: a good approximation for $\sin(x)$ is $x – \frac{1}{6}x^3 + \frac{1}{120}x^5 – …$, and a good approximation for $\cos(x)$ is $1 – \frac{1}{2}x^2+ \frac{1}{24}x^4 – …$. That means the fraction becomes:

$\frac{x\left(3 – \frac{1}{2}x^2 + \frac{1}{40}x^4 – …\right)}{3 – \frac{1}{2}x^2 + \frac{1}{24}x^4 – …}$

Ignoring the $x$, the rest of the numerator and denominator only differ in the $x^4$ term (and onward), which, for a small angle, makes for a very small error.

How does it help? Well, sine and cosine are circular functions, which means they can be measured off of a circle. Given a pair of perpendicular axes and a circle centred on them, any point on the circumference is $R \sin(\theta)$ above the horizontal axis and $R \cos(\theta)$ to the right of the vertical axis, assuming the circle has radius $R$ and the point forms an angle $\theta$ with the horizontal axis.

That means, if you construct an angle of, say, $\frac{\pi}{6}$, you should be able to construct and measure $\frac{3R \sin(\theta)}{2R + R\cos(\theta)}$, which is approximately $\theta$.

### The picture

A proper version. Chord AB has length 74mm, and the perpendicular ray CG has length 212mm. (Click to enlarge)

So, I drew that kind of circle. I constructed perpendiculars. I extended lines. I measured them. And I did some simplification.

If I wanted an estimate for π, I’d need to multiply everything by 6; meanwhile, I’d measured double $R\sin(\theta)$, the chord of the circle (AB in the diagram), so I’d need to multiply that number — 74mm — by 9 to get 666. Oooo!

On the bottom, I’d extended my horizontal chord (CA) by two radii to get $2R + R\cos(\theta)$, which I measured to be 212mm (CG). Then I did some tedious long division to get 3.14151, which isn’t bad for something knocked up on the sofa with a borrowed geometry kit. It’s almost too good to be true.

### Well, yes, of course I cheated

Estimates of π that are good to nearly five significant figures don’t pop off of such pages, at least, not without a great deal of preparation. For example, an unscrupulous geometer might fire up Desmos, draw the line $y = \pi x$, and see where it falls unusually close to a lattice point. (Better mathematicians than me would use a continued fraction to get the best estimate possible — although I did consider $\frac{355}{113}$, I couldn’t make it look natural. That was my first attempt.)

I found that $\frac{333}{106}$ was pretty much bang on the money — certainly, good enough for this. However, I needed the top to be a multiple of 18; it’s already a multiple of 9, so doubling it would work. I also know that $\sin\left(\frac {\pi}{6}\right) = \frac{1}{2}$, giving me a radius of $\frac{666}{18} = 74\text{mm}$.

From there, all that remained to do was fudge the horizontal chord marks ever so slightly so they coincided with the 212mm I’d pre-measured, and boom! A natural-looking, but ever-so-good estimate of π.

## About the author

• #### Colin Beveridge

Colin Beveridge is the author of Basic Maths For Dummies. Based in Weymouth, Dorset, he divides his time between persuading people that C4 is fun and trying to get his head around basic game theory.