You’ve probably seen it on a dusty chalkboard in a movie or stuck on a middle school classroom wall. $\pi \approx 3.14$. It’s everywhere. But honestly, the formula to find pi isn’t just one single thing. It’s a multi-thousand-year-old obsession that has moved from literal ropes and dirt to supercomputers crunching numbers for months on end.
Pi is a bit of a weirdo. It’s an irrational number, which basically means it never ends and never repeats a pattern. You can’t write it as a simple fraction like 1/3. Because of that, finding a way to calculate it precisely has become a sort of "Everest" for mathematicians. If you’re just trying to bake a pie or build a circular fire pit, $3.14$ or $22/7$ works fine. But if you’re NASA trying to land a rover on Mars, you need a lot more "oomph" in your decimal places.
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The Basic Formula to Find Pi (The Old School Way)
At its most fundamental level, pi is a ratio. That’s it. If you take any circle—doesn't matter if it's a penny or a galaxy—and measure the distance around it (the circumference) and then divide that by the distance across it (the diameter), you get pi.
The math looks like this:
$$\pi = \frac{C}{d}$$
It sounds easy. It’s not. The problem is that measuring a curved line with perfect accuracy is actually a nightmare. Ancient Egyptians and Babylonians knew this. They had rough estimates, like $3.125$ or $3.16$. They were close, but in the world of geometry, "close" still leaves you with a wobbly wheel.
Around 250 BCE, a guy named Archimedes got clever. He didn't try to measure a circle directly. Instead, he drew a polygon inside the circle and another one outside of it. By calculating the perimeters of these straight-edged shapes, he knew pi had to be somewhere in the middle. He started with hexagons and worked his way up to a 96-sided shape. He proved that pi was between $3 \frac{10}{71}$ and $3 \frac{1}{7}$. This "method of exhaustion" was the gold standard for nearly 2,000 years.
When Calculus Changed the Game
Everything got faster when we stopped drawing shapes and started using infinite series. This is where the formula to find pi gets really trippy. Imagine adding a bunch of fractions together forever. If you pick the right fractions, they eventually settle on a specific number.
Gottfried Wilhelm Leibniz and James Gregory discovered a famous one, though it’s painfully slow.
$$\pi = 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots \right)$$
If you use this formula, you’ll be at it for a long time. You’d need to add up about 300 terms just to get two measly decimal places right. It’s beautiful, sure, but it’s useless for real-world speed.
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Isaac Newton actually got so bored during the plague in 1665 (sound familiar?) that he calculated pi to 15 digits using a different fluxion method. He later wrote, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." Even geniuses get bored.
The Modern Heavy Hitters: Chudnovsky and Beyond
Fast forward to right now. We aren't using 96-sided polygons anymore. We use the Chudnovsky algorithm. If you look at it, it looks like a mess of square roots and massive integers. It’s what Google used in 2022 to calculate pi to 100 trillion digits.
$$\frac{1}{\pi} = 12 \sum_{k=0}^\infty \frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)! (k!)^3 (640320)^{3k + 3/2}}$$
Don't let the complexity scare you. Basically, this formula is an ultra-efficient way to squeeze out digits. Every time a computer runs one "loop" of this math, it gives us about 14 new digits of pi. That’s why we use it. It’s fast.
But why do we even need 100 trillion digits? Honestly, we don't.
For almost all physical applications in our universe, you only need about 39 or 40 digits. With 40 digits of pi, you could calculate the circumference of the observable universe with an error margin no larger than the width of a single hydrogen atom. Seriously. Anything beyond that is just a stress test for computers or a flex for mathematicians.
Common Mistakes People Make with Pi
Most people think $22/7$ is pi. It’s not. It’s an approximation.
$22/7 = 3.142857...$
$\pi = 3.141592...$
It’s close enough for a high school geometry quiz, but it’s technically "wrong." Another misconception is that because pi is infinite, every possible string of numbers must exist within it—like your phone number or the digital code for a Shakespeare play. While most mathematicians suspect this is true (calling it a "normal number"), it actually hasn't been proven yet. Pi could, theoretically, stop having 7s at some point, though it's highly unlikely.
How You Can Calculate Pi at Home (Buffon’s Needle)
If you want to feel like a mad scientist, you can find pi using a box of needles or toothpicks. This is called Buffon’s Needle problem.
- Drop a bunch of needles on a sheet of lined paper.
- The distance between the lines should be the same as the length of the needle.
- Count how many needles cross a line.
- Use this formula: $2 \times (\text{total needles} / \text{crossing needles})$.
It’s a "formula to find pi" based on probability. It’s wild because it links circles to straight lines and random chance. The more needles you drop, the closer you get to $3.14$.
Actionable Steps for Exploring Pi
If you're looking to actually use this or dive deeper, here's how to move forward:
- For coding practice: Try writing a Python script using the Bailey–Borwein–Plouffe (BBP) formula. It's unique because it allows you to calculate the $n^{th}$ digit of pi without calculating all the digits before it.
- For precision work: If you are doing CAD (Computer-Aided Design) or engineering, use the built-in
Math.PIconstant in your programming language rather than typing $3.14$. Most languages use the IEEE 754 double-precision format, which gives you about 15-17 significant digits. - For the "Cool Factor": Check out the Pi Search Engine. You can type in your birthday and see exactly where it appears in the first 200 million digits of pi.
- Verify your math: If you're manually calculating circles, always keep the $\pi$ symbol in your equations until the very last step. This prevents "rounding error creep," where small mistakes at the start of a problem turn into huge errors by the end.
Pi isn't just a number; it's a window into how the universe is put together. Whether you use the simple $C/d$ or a massive algorithm, you're tapping into a logic that's been around since long before humans started counting.