Estimating the value of Pi

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I am trying to estimate the value of pi by checking how often a pair of random variables are inside a circle

I will draw many more random numbers (1,000,000 or more) and calculate pi as the fraction of points inside the circle times the area of the box.

The area of the box is 2x2=4, and a pair is inside or on the circle if x^2+y^2≤1 .

To start, how can i plot a circle x^2 + y^2 = 1 ?


Here are three examples of how to calculate π using the rand method. The first two use generators, while the last one is an ordinary loop. Notice that I avoid creating an array, in order to avoid memory allocations.

pisum1(N) = count(true for _ in 1:N if rand()^2 + rand()^2 <= 1) * 4/N

pisum2(N) = count(rand()^2 + rand()^2 <= 1 for _ in 1:N) * 4/N

function pisum3(N)
    s = 0
    for _ in 1:N
        s += (rand()^2 + rand()^2 <= 1)
    end
    return 4s/N
end

Let's test how fast they are:

julia> using BenchmarkTools

julia> N = 10^7
10000000

julia> @btime pisum1($N)
  105.221 ms (0 allocations: 0 bytes)
3.1410964

julia> @btime pisum2($N)
  81.046 ms (0 allocations: 0 bytes)
3.1416524

julia> @btime pisum3($N)
  34.942 ms (0 allocations: 0 bytes)
3.141756

If you want performance, you should go with the straight loop.

What Is Pi, and How Did It Originate?, For Pi Day 2016 I tried to calculate π by hand, using an infinite series. It goes ok. Before you even Duration: 18:40 Posted: Mar 13, 2016 One method to estimate the value of \(\pi \) (3.141592) is by using a Monte Carlo method. In the demo above, we have a circle of radius 0.5, enclosed by a 1 × 1 square. The area of the circle is \(\pi r^2 = \pi / 4 \), the area of the square is 1. If we divide the area of the circle, by the area of the square we get \(\pi / 4 \).


You can plot a circle in 1 line with Plots.jl:

using Plots
gr()

plot(cos, sin, 0, 2pi, line=4, leg=false, fill=(0,:orange), aspect_ratio=1)

This plots cos and sin components (x and y coordinates) between 0 and 2pi and then fills the area between the curves. Fixing the aspect ratio makes sure that the circle looks like a circle. This is the resulting plot:

Thanks to DNF for pointing out an easy solution.

The area of a unit circle is Pi and so the area of the polygon can be calculate to find an Duration: 3:15 Posted: Jun 26, 2019 An easy way to estimate the value of pi is to divide a circle's circumference by its diameter. Measure the circumference of a cylinder or circle using a thin piece of string. (The circumference is the distance around the circle.)


You don't need to plot anything if you just want to estimate pi. Generate a n by 2 matrix with every element a normal. Then you can loop through the rows to see if the sum of the squares are smaller than 1.

Estimating the value of Pi using Monte Carlo. Monte Carlo estimation. Monte Carlo methods are a broad class of computational algorithms that rely on repeated  Pi, (π), is used in a number of math equations related to circles, including calculating the area, circumference, etc. and is widely used in geometry, trigonometry and physics. This app estimates the value of pi by comparing the area of a square and an inscribed circle.


How to estimate a value of Pi using the Monte Carlo method - generate a large number of random points and see how many fall in the circle enclosed by the unit​  You can find the value of pi with a mass and a spring. The value of pi is related to the local gravitational field. You can find the value of pi using random numbers (this one is my favorite).


Explore this Article. Calculating Pi Using the Measurements of a Circle. Calculating Pi Using an  Estimating the value of Pi using Monte Carlo 1. Initialize circle_points, square_points and interval to 0. 2. Generate random point x. 3. Generate random point y. 4. Calculate d = x*x + y*y. 5. If d <= 1, increment circle_points. 6. Increment square_points. 7. Increment interval. 8. If


This means you need an approximate value for Pi. polygon and use this to calculate the perimeter and diameter as an estimate for Pi. That means that the circumference is equal to 2 * Pi, so half a circle or 180 degrees equals Pi (usually said to be Pi radians). It turns out to be fairly easily provable in Calculus that the Derivative of the Inverse Tangent (a trigonometry term) is equal to 1/(1 + X 2 ).