General formulas vs. special cases with τ and π

An enlightening discussion about pi and tau.

General formulas vs. special cases with τ and π

by 1=0 » Fri Mar 31, 2017 9:44 am

In basic geometry, there are four main formulas involving the circle constant.
Circumference of a circle: C=τr=2πr
Area of a circle: A=1/2τr^2=πr^2
Surface area of a sphere: S=2τr^2=4πr^2
Volume of a sphere: V=2/3τr^3=4/3πr^3
These formulas seem to be equally simple with tau or pi. However, they are all special cases of more general formulas.
Circumference is a special case of arc length s=θr
Circular area is a special case of area of a sector: A=1/2θr^2
Spherical area is a special case of area of a spherical lune: S=2θr^2
Volume of a sphere is a special case of volume of a spherical wedge: V=2/3θr^3
When comparing the special cases to the more general formulas, we see an obvious win for tau:
C=τr is consistent with s=θr; C=2πr is not.
A=1/2τr^2 is consistent with A=1/2θr^2; A=πr^2 is not.
S=2τr^2 is consistent with S=2θr^2; S=4πr^2 is not.
V=2/3τr^3 is consistent with V=2/3θr^3; V=4/3πr^3 is not.
This not only shows tau to be the more natural constant; it also shows that it is more convenient in geometry. Rather than memorizing two sets of four formulas, you only have to memorize one. It is also impossible to get the two sets mixed up because they will be exactly the same with tau, but with pi they are different by a factor of two. This, in addition to the fact that the circumference is the most important of those formulas, shows why tau is obviously the best constant to use.
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