θ : 0 → τ
One full
turn is τ.
Spin a radius all the way around a circle and the angle you cover is τ = 2π ≈ 6.283 radians. Not π — π is only halfway. τ is the whole turn.
θ0.000
0% of a turn
θ : 0 → τ
Spin a radius all the way around a circle and the angle you cover is τ = 2π ≈ 6.283 radians. Not π — π is only halfway. τ is the whole turn.
What one sweep gives you
As the radius r sweeps through an angle θ, its tip draws an arc of length r·θ. Carry it through a full turn and you've traced the whole way around.
The wedge swept out so far has area ½r²θ. Let the radius come all the way back to where it started and the wedge becomes the entire disc.
Slicing a circle into thin wedges is just this sweep, cut into steps.
Each sliver is a tiny sweep dθ; stack every sliver from 0 to τ and you have the whole circle back — the continuous twin of the slice-and-rearrange proof.