Take any circle in the world, a coin, a pizza or the orbit of a satellite, and divide the distance around it by the distance across it. You always get the same number: 3.14159 and then digits that keep going forever without settling into a pattern. That number is pi, and people have been chasing its digits for more than 2,000 years.
So what is pi, why does it never end, and how many of those digits do you actually need? This guide answers all three, with worked examples you can use in math class.
Quick answer
Pi (π) is the ratio of a circle’s circumference to its diameter, about 3.14159. It is the same for every circle of any size. Pi never ends because it is an irrational number: it cannot be written as a fraction of two whole numbers, so its decimal digits go on forever without repeating. Johann Lambert proved this in the 1760s. In practice you rarely need many digits. School problems use 3.14 or 22/7, and NASA’s Jet Propulsion Laboratory uses about 15 decimal places for interplanetary navigation.
Key takeaways
- Definition: π = circumference ÷ diameter, for any circle.
- Two formulas: circumference C = πd (or 2πr) and area A = πr².
- Irrational: no fraction equals pi exactly, so its digits never end or repeat.
- Close enough: 3.14 and 22/7 are approximations, not the true value.
- Pi Day is March 14 (3/14).
What Is Pi?
Pi is a constant, a number that never changes. Every circle has two key lengths: its circumference (the distance around the edge) and its diameter (the distance straight across, through the center). Divide the first by the second and the answer is always pi.
The first 30 decimal places are:
3.141592653589793238462643383279…
The Greek letter π stands for it because it is the first letter of the Greek word for “perimeter.” The Welsh mathematician William Jones first used the symbol this way in 1706, and the Swiss mathematician Leonhard Euler made it popular a few decades later.
The Two Pi Formulas You Need
| To find | Formula | Remember |
|---|---|---|
| Circumference | C = πd or C = 2πr | Diameter is twice the radius |
| Area | A = πr² | Square the radius first, then multiply by π |
Worked example 1: a bike wheel
- A bike wheel has a diameter of 26 inches.
- C = π × 26 ≈ 3.14159 × 26 ≈ 81.7 inches.
- Every full turn of the wheel moves the bike about 81.7 inches, a little under 7 feet.
Worked example 2: the pizza question
Which gives you more pizza: one 16-inch pizza or two 10-inch pizzas? Pizza sizes are diameters, so halve them to get the radius.
- One 16-inch pizza: radius 8, so A = π × 8² = π × 64 ≈ 201 square inches.
- Two 10-inch pizzas: radius 5, so A = 2 × π × 5² = 2 × π × 25 ≈ 157 square inches.
- The single large pizza wins by about 44 square inches, because area grows with the square of the radius.
Check your answers with our free Scientific Calculator, which has a π key.
Why Does Pi Never End?
Numbers come in two broad kinds. Rational numbers can be written as a fraction of two whole numbers, like 1/4 or 2/3. Their decimals either stop (1/4 = 0.25) or fall into a repeating pattern (2/3 = 0.666…).
Irrational numbers cannot be written as any such fraction. Their decimals go on forever and never repeat. The square root of 2 is one example, and pi is another.
In 1761 the Swiss mathematician Johann Heinrich Lambert proved that pi is irrational. That proof is why we know, for certain, that no amount of computing will ever reach a last digit or find a repeating loop. It is not that nobody has found the end yet; there is no end to find.
In 1882 the German mathematician Ferdinand von Lindemann went further. He proved pi is also transcendental: it is not the answer to any ordinary polynomial equation with whole-number coefficients. That settled an ancient puzzle called “squaring the circle.” Using only a compass and straightedge, it is impossible to draw a square with exactly the same area as a given circle.
How People Calculated Pi Through History
| When | Who | Result |
|---|---|---|
| About 1650 BCE | Egyptian Rhind Papyrus | About 3.16 |
| About 250 BCE | Archimedes of Syracuse | Between 3 10/71 and 3 1/7 |
| About 480 CE | Zu Chongzhi, China | 355/113 (correct to 6 decimals) |
| 1706 | John Machin | 100 decimal places by hand |
| 2020s | Supercomputers and data-storage teams | Over 100 trillion digits |
Archimedes’ method is still taught today. He drew polygons inside and outside a circle and measured their perimeters. With 96-sided polygons, he squeezed pi between two values. More sides meant a tighter squeeze.
How Many Digits of Pi Do You Really Need?
Far fewer than you might think. Engineers at NASA’s Jet Propulsion Laboratory use 3.141592653589793, which is 15 decimal places, to steer spacecraft across the solar system. With that value, a circle with a radius of 12.5 billion miles (about Voyager 1’s distance from Earth when JPL made the comparison) comes out wrong by only about an inch and a half.
The same team has also pointed out that about 40 decimal places would be enough to calculate the circumference of the entire visible universe to within the width of a single hydrogen atom. Records of trillions of digits are mainly used to test computer hardware and software.
Common Mix-Ups About Pi
Myth: Pi equals 22/7. 22/7 is about 3.142857, which is close but wrong from the third decimal place on. No fraction equals pi exactly.
Myth: Pi is exactly 3.14. 3.14 is a rounded value for quick calculations. Use the π key or 3.14159 when your teacher asks for more accuracy.
Myth: Scientists just haven’t found the last digit. Lambert’s proof shows there is no last digit. The digits truly go on forever.
Myth: Pi contains every possible number sequence. That would be true if pi is a “normal” number, but nobody has proved it. Mathematicians suspect it, and that is all.
Frequently Asked Questions
What is pi to 10 decimal places?
3.1415926535. Rounded to two decimal places, the value most homework uses, it is 3.14.
Why is pi important?
Pi appears anywhere circles, spheres, waves or rotation show up: engineering, physics, GPS, electronics, sound and statistics. Without it you could not calculate the size of a pipe, a planet’s orbit or a radio signal.
Who discovered pi?
No single person. Ancient Babylonians and Egyptians used approximations, Archimedes found a method to calculate it around 250 BCE, and William Jones gave it the symbol π in 1706.
When is Pi Day?
March 14 (3/14 in US date format). The US House of Representatives recognized it in 2009, and it also happens to be Albert Einstein’s birthday.
Is pi a rational or irrational number?
Pi is irrational. It cannot be written as a fraction of two whole numbers, and its decimal expansion never ends or repeats.
Quick Recap
- Pi is circumference divided by diameter, about 3.14159, for every circle.
- C = πd and A = πr² are the two formulas to memorize.
- Pi is irrational (proved in 1761), so its digits never end or repeat.
- NASA’s JPL needs only about 15 decimal places to fly spacecraft.
NASA’s Jet Propulsion Laboratory explains how many pi decimals NASA uses, and Wolfram MathWorld collects the formulas in its reference entry on pi. See more on our mathematics page, read why the calendar needs leap years, then practice with the free Math Quiz.
