τ (tau) equals 2π ≈ 6.28318. Its defining property is simple: one full revolution of a circle is exactly τ radians. Half a turn is τ/2 = π radians. A quarter turn is τ/4. For those who find this more natural than π, the circle constant is τ, not π.
One full revolution = τ radians. τ/4 = 90°. τ/2 = 180° = π radians. The circumference of a circle is C = τr.
The case for τ: the circumference formula becomes C = τr (circumference = tau × radius), and any fraction of a turn is that fraction times τ. sin(τ) = 0, cos(τ) = 1 (returning to start). Euler's identity in terms of τ: e^(iτ) = 1, a full rotation. The case against: π is established in every textbook and formula for centuries.
Comparison of formulas using tau vs pi
| Formula | with π | with τ |
|---|---|---|
| Circumference | 2πr | τr |
| Area of circle | πr² | τr²/2 |
| Full turn | 2π rad | τ rad |
| Euler identity | eⁱπ+1=0 | eⁱτ=1 |
| Gaussian integral | √(2π) | √τ |
τ = 2π is transcendental (since π is transcendental). Whether it is the better circle constant is a matter of taste, not mathematics. The Tau Manifesto (Michael Hartl, 2010) makes the pedagogical argument. τ to 20 digits: 6.28318530717958647692…
With π, a quarter turn is π/2: half of the full-turn constant. With τ, a quarter turn is τ/4: literally one quarter. Every fraction of a turn maps directly to the same fraction of τ.
Tau is exactly 2 times pi, approximately 6.28318530717958647692. It is irrational and transcendental. One tau radian equals one full circle, making it arguably more natural than pi as the circle constant. Proposed by Bob Palais in 2001 and popularised by Michael Hartl's Tau Manifesto. Tau Day is June 28 (6.28). Euler's identity with tau reads e^(iτ) = 1: a full rotation of the complex plane returns to the start.
Tau τ is irrational. Its decimal expansion never ends and never repeats. Every digit shown below is computed from the circle definition.