Approximating Pi

Approximating Pi

I programmed three ways of approximating Pi. I made the Gregory-Leibniz Series, the Nilakhanta Series, and I did it using geometry.

Gregory-Leibniz Series

This method is seen as one of the most famous and most beautiful way to approximate Pi. The only thing which isn't really cool about this calculation, is that it takes 300 terms to get only two decimals correct.

The formula is as follows: (π / 4) = 1 - (1 / 3) + (1 / 5) - (1 / 7) + ...

I made a for loop which looks like this: for(var i = 3; i < 2000; i += 2). I used 3 because of the fractions. Underneath, I first put the approximation and then Math.PI.

Nilakhanta Series

The special thing about this method is that it is developed in the 15^th century. This method converges more quickly as well. That means it gets quicker to the correct answer.

The Nilakhanta Series looks like this: π = 3 + (4 / (2*3*4)) - (4 / (4*5*6)) + (4 / (6*7*8)) + ...

After about 500 iterations, the approximation looks like this:

Geometry

Note: This idea is from The Coding Train, and I decided to make this in JavaScript. (https://youtu.be/5cNnf_7e92Q)

Approximating Pi using geometry requires a little bit of maths. As you probably know, the circumference of a circle is 2πr, and the area is πr². Now, if you draw a perfect square which touches the edges of the circle, the lengths of the square edges is 2r, which makes the area 2r * 2r = 4r².

We can use these details to approximate Pi! We can draw pixels, either within the circle, or outside the circle but inside the square. Then we can approximate Pi with the following formula: π = 4 * (dotsInCircle / totalDots)

I made this geometric way of approximating Pi using the canvas below. The red dots beling in the circle, the blue dots don't.

Number of pixels per frame (the bigger the number, the faster it goes, but that will take a lot of your device as well)

As you can see with this last method, it gets more accurate as more pixels are being drawn.