December 23, 2019

The Monte Carlo Method

This is my solution to dailycodingproblem.com problem #14.

The Problem

The area of a circle is defined as πr^2. Estimate π to 3 decimal places using a Monte Carlo method.

Hint: The basic equation of a circle is x2 + y2 = r2.

My Solution

For this challenge I am going to be using the cli-graph library to draw to the console.

Let’s imagine a circle with a radius of 10. Because we don’t know what Pi is, we can’t draw a circle yet… but we can create a square graph to hold the circle.

const Graph = require('cli-graph')

const RADIUS = 10

const graph = new Graph({
	height: RADIUS * 2,
	width: RADIUS * 2,


graph a

Plotting a point

The way we are going to estimate Pi, is by plotting millions of random points on the graph, and finding the ones that are inside our circle. We can use the ratio of points inside the circle to those outside as a rough estimate of the area of the circle.

Let’s start with plotting a single point:

graph.addPoint(5, 5)

graph b

Calculating the radius

Now we have a point on the graph, we need to che check if that point is inside our imaginary circle.

To do this, we can use equation of a circle supplied to find the radius of a circle that contains a particular point on it’s circumference.

If that radius is less or equal to our RADIUS, than we know that point is inside our circle!

Equation of a Circle:

Math.pow(x, 2) + Math.pow(y, 2) === Math.pow(radius, 2)

Let’s twist it around to get the radius:

radius === Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2))

Using these equations we can create a function to calculate the radius of a given point on the graph.

const calculateRadius = (x, y) => Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2))

calculateRadius(5, 5)
// 7.071...

And use that function to check if a point is within a given circles radius.

const isPointInCircle = (x, y, r) => calculateRadius(x, y) <= r

isPointInCircle(5, 5, RADIUS)
// true, 5,5 is within our circle

isPointInCircle(10, 10, RADIUS)
// false, 10,10 is not within our circle

Let’s get a random point

We are also will return a number between -RADIUS and RADIUS

const getRandomPoint = (r) => Math.random() * (r * 2) - r

We can now pick a random coordinate on our graph, and check if it is inside a circle:


Now we can try a bunch of random points within our square, and plot if they are inside or outside our circle.

for (let i = 0; i < 10000; i++) {
	const x = getRandomPoint(RADIUS)
	const y = getRandomPoint(RADIUS)

  const char = isPointInCircle(x, y, RADIUS)
    ? '•'
    : ' '

	graph.addPoint(x, y, char)


graph c

We can see our circle! While it isn’t a perfect circle, we can still use it to approximate Pi.

The area of a circle is defined as πr2.

area === Math.PI * Math.pow(r, 2)

We can twist this around:

Math.PI === area / Math.pow(r, 2)

But how do we get the area of our circle? What we can do, is find the rough percentage of space that our circle occupies in the square

const count = 10_000_000 // ten million is usually good enough for 3 decimal places

let hits = 0

for (let i = 0; i < count; i++) {
  const x = getRandomPoint(RADIUS)
  const y = getRandomPoint(RADIUS)
  if (isPointInCircle(x, y, RADIUS)) {
    hits += 1

const approximateCircleRatio = hits / count
// ~78.54%

We can also calculate the area of our graph:

const areaOfGraph = (RADIUS * 2) * (RADIUS * 2)

// > 400

By combining the approximateCircleRatio and the areaOfGraph we can find the approximate area of a circle.

const approximateAreaOfCircle = approximateCircleRatio * areaOfGraph

And now we can use the area of a circle to to approximate Pi!

const approximatePi = approximateAreaOfCircle / Math.pow(RADIUS, 2)

console.log(`Approximate PI: ${approximatePi.toFixed(3)}`)
// Approximate PI: 3.142

The more points we plot on the graph, the more accurate our estimate will be, but it will also take more time.

© 2020 George Czabania