22/7 — Pi's Famous Fraction
Somewhere in school, almost everyone learns that pi is "about 22/7." It's a genuinely useful approximation — and also a common source of a very stubborn misconception.
What 22/7 actually is
The fraction 22/7 is a simple rational approximation of pi. Do the division and you get:
22 ÷ 7 = 3.142857142857142857…
Notice that the block 142857 repeats forever. That repetition is your first clue that something is off, because the real pi never settles into a repeating pattern. Compared side by side:
- 22/7 =
3.142857142857… - π =
3.141592653589…
They agree on 3.14 and then part ways at the third decimal.
Exactly how accurate is it?
22/7 is correct to two decimal places — it nails 3.14 and no further. The error is small but real:
- The gap is
22/7 − π ≈ 0.00126. - As a percentage, that's an overshoot of roughly 0.04%.
- 22/7 is larger than pi, not smaller.
For a two-character fraction, that's remarkably good. To put 0.04% in perspective: if you used 22/7 to compute the circumference of a circle the size of a football field, you'd be off by less than the width of your hand.
Is 22/7 equal to pi? No.
This is the important part. 22/7 is not pi, and it never can be. Pi is irrational, which means it cannot be written as one whole number divided by another. Its decimals go on forever without ever repeating. 22/7, by contrast, is a ratio of two integers by definition, so its decimals do repeat — that endless 142857 is proof it's a different animal entirely. Calling 22/7 "pi" is like calling a photograph of a person that person. It's a very good likeness, and it is not the thing itself.
Archimedes and the polygon bounds
22/7 isn't a modern shortcut — it's more than two thousand years old. Around 250 BC, Archimedes of Syracuse pinned pi down by drawing polygons inside and outside a circle and measuring their perimeters. Using 96-sided polygons, he proved that pi sits between two fractions:
223/71 < π < 22/7
In decimals that's 3.1408… < π < 3.1428…. So 22/7 is precisely Archimedes' upper bound for pi — a guaranteed ceiling, established by rigorous geometry long before anyone had decimals to work with.
Better fractions exist
If you want more accuracy without giving up on fractions, there's a famous one: 355/113. It equals 3.1415929… and is correct to six decimal places — an astonishing return for such small numbers. It was found by the Chinese mathematician Zu Chongzhi in the fifth century, and is sometimes called Milü ("the detailed ratio"). It's easy to remember, too: write the odd digits in doubled pairs, 11-33-55, and split them as 113 into 355.
Pi Approximation Day
Because 22/7 is so beloved, it has its own holiday. In the day/month date format used across much of the world, July 22 is written 22/7 — so July 22 is celebrated as Pi Approximation Day, a lighthearted companion to the March 14 (3/14) Pi Day.
When 22/7 is good enough — and when it isn't
Use 22/7 freely for mental math, homework sanity checks, and any everyday estimate where two decimals of accuracy is plenty. It's fast, memorable, and lets you cancel the 7 cleanly when a radius is a multiple of 7. Reach for something better — the full decimal value, or 355/113 — when you're doing engineering, science, or anything where a 0.04% error would compound. And if you need pi's actual digits, that's a different question with a much longer answer.