Monte Carlo Estimator: Estimating π and Beyond

TL;DR

Built a Monte Carlo simulation in C++ to estimate π using random sampling and statistical convergence. Demonstrates how randomness can solve deterministic problems — essential for scientific computing and game development.

Who This Is For

Game developers implementing procedural generation
Students learning probability and simulation
Anyone curious about computational mathematics

Prerequisites: Basic C++ knowledge, understanding of loops and random numbers, high school geometry helpful.

Monte Carlo methods have always fascinated me — especially the idea that randomness can be harnessed to estimate something as fundamental as π. So I decided to build a Monte Carlo Estimator in C++ to see this concept in action.

This was a simple but eye-opening project that deepened my understanding of probability, convergence, and the elegance of using simulation to solve problems that look purely theoretical on paper.

Project Overview

A C++ program that uses Monte Carlo simulation to estimate π by randomly generating points and calculating the ratio of points inside a circle to total points.

🎯 What the Program Does

Here's the basic idea:

Randomly generate points inside a square.
Check whether each point falls within a unit circle centered at the origin.
Estimate π using the formula:
π ≈ 4 × (points in circle / total points)
Repeat this process thousands (or millions) of times.
Display the results as the approximation gets closer to the real value.

It's almost magical watching randomness settle into predictability.

The Math Behind It

The area of a circle is πr², and the area of a square is (2r)² = 4r². So the ratio of circle area to square area is π/4, which gives us π = 4 × (circle points / total points).

🧠 What I Learned

1. Randomness in Practice

I dug into how C++ generates random numbers with rand() and srand(). Even though they're fairly basic, they were perfect for demonstrating the core idea behind Monte Carlo simulations.

// Example of random point generation
srand(time(0));
for (int i = 0; i < iterations; i++) {
    double x = (double)rand() / RAND_MAX * 2 - 1; // -1 to 1
    double y = (double)rand() / RAND_MAX * 2 - 1; // -1 to 1

    if (x*x + y*y <= 1) {
        pointsInCircle++;
    }
}

Environment Used

Last tested with: GCC 13.x, C++17 standard, tested on Windows/Linux/macOS

Limitations & Trade-offs

Pseudo-randomness: rand() is not cryptographically secure - Convergence speed: Slow convergence rate compared to analytical methods - Memory usage: Scales linearly with iteration count - Accuracy ceiling: Limited by floating-point precision

2. Geometric Probability

This project reinforced how geometry and probability intersect. The fact that you can approximate π by throwing darts at a virtual square is one of the most elegant examples of applied math I've seen.

3. Balancing Accuracy and Performance

I experimented with different iteration counts to see how quickly the estimates converged. More samples definitely mean better accuracy, but they also mean longer runtime, which was an interesting tradeoff to observe firsthand.

Performance Trade-off

More iterations = better accuracy, but also longer runtime. I found that 1 million iterations typically give a good balance of accuracy and speed.

4. Visualizing Progress in the Console

I added print statements to show the estimation improving over time. Watching the number stabilize closer and closer to 3.14159 made the learning feel real and tangible.

📷 Screenshot

📂 Tech Stack

C++
Math & Geometry
Terminal-Based UI

🔗 Check It Out

📁

Source Code

It's easy to underestimate small projects like this, but they can teach you a lot. This Monte Carlo Estimator wasn't just a way to practice C++ — it was a reminder that sometimes the simplest ideas can reveal the most about how math and programming work together.

What's Next

I'm considering expanding this to estimate other mathematical constants or explore different geometric shapes and their properties.

If you're curious about Monte Carlo methods or just want to see the code in action, check out the repo or drop me a message — I'd love to hear your thoughts.

— Maruf