Is the value of Pi wrong?

Answer 1

The value of pi (π) is wrong in the sense that it's an unnatural choice for a circle constant. The circle constant used by Archimedes, π is defined as the circumference of a circle divided by the diameter (C/D = 3.1415...), so there are 2π radians per circle. So it's really the "semicircle constant", not the circle constant. He probably chose to divide by diameter because it's easier to measure, but circles are mathematically defined by their radius, not their diameter.

A more natural circle constant is the circumference divided by the radius (C/r = 6.283185...). This was the value used by Al-Kashi, and is called tau (τ) by people who want to promote it as the circle constant to be taught in schools and used in calculations. There are τ (=2π) radians per circle.

(In fact, the first use of the symbol π was by David Gregory in 1697 to represent a constant of circumference / radius, like τ. However, it was written as a fraction π∕ρ. π originally just meant "circumference"!)

Tau is a better choice because it makes distance around the circle intuitive. 1/4 of the way around a circle is τ/4 radians. 7/8 of the way around is 7τ/8 radians, etc.

It also makes it more obvious that the area of a circle is the result of an integral. For instance, the area of a polygon is 1/2 ⋅ perimeter ∙ apothem. If you increase the number of sides to infinity, it becomes a circle, with an area of 1/2⋅τr⋅r = 1/2⋅τr².

Read the related links for more examples of why τ makes more sense as the circle constant.