Tau (mathematical constant)

This article is about the number. For the Greek letter, see Tau (letter).

Tau (τ) (/ˈtɔː, ˈtaʊ/) is a mathematical constant that is the ratio of a circle's circumference to its radius. This produces a number, and that number is always the same. However, the number is rather strange. The number starts as 6.283185307179586 and continues without end. Numbers like this are called irrational numbers.

Tau is an endless string of numbers

The radius is the shortest connection between the center and the circumference of a circle. The circumference is the distance around a circle. Even though the radius and circumference are different for different circles, the number tau remains constant: its value never changes. This is because the relationship between the circumference and radius is always the same.

Fundamentals

Definition

τ is commonly defined as the ratio of a circle's circumference c to its radius r:

$\tau ={\frac {c}{r}}$

Approximate value

Tau is often written as τ. It is also an irrational number, meaning it cannot be written as a fraction (${\frac {a}{b}}$), where 'a' and 'b' are integers (whole numbers). This basically means that the digits of tau that are to the right of the decimal go forever—without repeating in a pattern, and that it is impossible to write the exact value of tau as a number. Tau can only be approximated, or measured to a value that is close enough for practical purposes.

A value close to tau is 6.283185307179586476925287. Computers can be used to get better approximations of tau.

Tau in real life

There are different ways to calculate many digits of tau. This is of limited use though.

Tau can sometimes be used to work out the area or the circumference of any circle. To find the circumference of a circle, use the formula τ × r (Radius). To find the area of a circle, use the formula τ / 2 × r² (radius squared). This formula is sometimes written as $A={\frac {\tau r^{2}}{2}}$, where A is the variable for the area.

People generally celebrate June 28 as Tau Day, because June 28 is also written as 6/28, which represents the first three numbers 6.28 in the approximation of tau.

Tau in programming languages

This table lists some programming languages that have Tau, the circle constant for converting between turns and radians. All of these programming languages use the name "Tau" in some casing, but Processing also has "TWO_PI" and Raku also has the symbol "τ" for the same value.

Language	Identifiers	First Version	Year Released
C# / .NET	System.Math.Tau and System.MathF.Tau	5.0	2020
Crystal	TAU	0.36.0	2021
Eiffel	math_constants.Tau	Curtiss	Not yet released
GDScript	TAU	Godot 3.0	2018
Java	Math.TAU	19	2022
Nim	TAU	0.14.0	2016
Processing	TAU and TWO_PI	2.0	2013
Python	math.tau	3.6	2016
Raku	tau and τ
Rust	std::f64::consts::TAU	1.47.0	2020
Zig	std.math.tau	0.6.0	2019

Related pages

• π (mathematical constant)
• List of numbers
• Irrational numbers
• Square root of 2
• Golden ratio
• e (mathematical constant)
• Squaring the circle
• Transcendental number