There is no circle calculation without π. The number appears in every formula for circumference, area, spheres and cylinders. But what exactly does it mean, how do we know its value – and how many decimal places do you really need?
What π means
π is the ratio of a circle's circumference to its diameter:
The remarkable part: this ratio is the same for every circle, whether coin, cartwheel or the Earth's equator. Wrap a string once around a tin and measure it – it is a little more than three times the tin's diameter. Exactly 3.14159… times.
A short history of π
Around 4,000 years ago the Babylonians used 3 1/8 = 3.125 and the Egyptians roughly 3.16. Archimedes of Syracuse took a systematic approach around 250 BC: he drew polygons with up to 96 sides inside and outside a circle. The circle's circumference must lie between the two polygon perimeters. That gave him:
That is 3.1408 to 3.1429, and it is where the approximation 22/7 comes from. In the 5th century the Chinese mathematician Zu Chongzhi found the fraction 355/113, which is correct to six decimal places. Around 1600 Ludolph van Ceulen computed 35 decimal places, which is why π was long called the “Ludolphine number” in German. Today computers know trillions of digits.
π is irrational and transcendental
In 1761 Johann Heinrich Lambert proved that π is irrational: it cannot be written as a fraction of two whole numbers, and its decimals never end or repeat. In 1882 Ferdinand von Lindemann showed that π is even transcendental, i.e. not a solution of any polynomial equation with integer coefficients. That settled a problem more than 2,000 years old: “squaring the circle” – constructing a square of the same area with compass and straightedge – is impossible.
How precise do you need to be?
For most tasks a few digits are enough. What matters is the relative error, because it carries straight through to circumference and area:
| Approximation | Value | Deviation | Error at r = 1 m (area) |
|---|---|---|---|
| 3 | 3.0000000 | −4.51% | −1,416 cm² |
| 3.14 | 3.1400000 | −0.051% | −15.9 cm² |
| 22/7 | 3.1428571 | +0.040% | +12.6 cm² |
| 3.1416 | 3.1416000 | +0.0002% | +0.07 cm² |
| 355/113 | 3.1415929 | +0.00001% | +0.003 cm² |
With 3.14 you are off by about half a per mille. For homework, garden planning and trade work that is almost always enough. Mechanical engineering and surveying use the full value, which calculators and spreadsheets provide anyway. Even navigating space probes through the solar system needs only about 15 decimal places – as many as a computer stores in an ordinary floating-point number, and what the circle calculator uses internally.
When to round
Round π and every intermediate result as late as possible. If you first round the radius to two places, then multiply by 3.14 and then square, you collect several small errors. Use the π key on your calculator and round only the final result to the precision you need.
A memory aid for the first digits
A classic English mnemonic counts the letters of each word: “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.” The word lengths 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9 give 3.14159265358979. 14 March is celebrated as Pi Day because it is written 3/14 in US date format; since 2020 it has also officially been the International Day of Mathematics.
π in other formulas
π turns up wherever circles or rotation are involved: in the volume of a cylinder (π × r² × h), the surface of a sphere (4 × π × r²), in radians (180° = π) and even in statistics, in the normal distribution. How to calculate with π in radians is shown in the article Sector and radians.