Tau — Definition, Formula & Examples
Tau (τ) is a Greek letter used throughout mathematics to represent various quantities, most notably the circle constant τ = 2π ≈ 6.2832, which equals the ratio of a circle's circumference to its radius.
The lowercase Greek letter τ (tau) serves as a mathematical symbol in multiple contexts. In its most prominent modern usage, τ denotes the constant 2π, defined as the ratio C/r where C is a circle's circumference and r is its radius. Tau also appears as a variable in physics (e.g., torque, time constants) and statistics (e.g., Kendall's tau).
Key Formula
Where:
- = The circle constant tau, equal to the ratio of a circle's circumference to its radius
- = Pi, the ratio of a circle's circumference to its diameter, approximately 3.14159
How It Works
When τ represents the circle constant, you can substitute τ = 2π in any formula. A full rotation around a circle equals τ radians instead of 2π radians, which some mathematicians argue is more intuitive. For example, a quarter turn is τ/4 (instead of π/2), and a half turn is τ/2 (instead of π). Proponents of τ say it simplifies formulas because most occurrences of π in mathematics actually involve 2π.
Worked Example
Problem: Convert the angle 3τ/4 radians to degrees.
Step 1: Replace τ with 2π.
Step 2: Convert from radians to degrees by multiplying by 180°/π.
Answer: 3τ/4 radians = 270°, which is three-quarters of a full turn.
Why It Matters
Understanding τ helps you read modern math resources and some programming languages where τ is a built-in constant. In trigonometry and precalculus, recognizing that τ = 2π lets you quickly convert between the two notations when working with radian measure and periodic functions.
Common Mistakes
Mistake: Confusing τ with π and using 3.14 for τ.
Correction: Remember that τ = 2π ≈ 6.28, not 3.14. Tau is twice the value of pi.