The area of a circle, for example, would be r^2 pi/2.
This is actually my favorite example of why pi is wrong—it's the "exception" that proves the rule. To see why, set τ = C/r = 2 pi, and then consider the following chart of common quadratic forms:
integral of u 1/2 u^2
kinetic energy 1/2 m v^2
distance fallen 1/2 g t^2
spring energy 1/2 k x^2
triangular area 1/2 b h
circular area 1/2 τ r^2
We see that, far from causing "different complications", using the right circle constant brings the area of a circle into a more natural form. The 1/2 in the formula for circular area is actually a missing factor; using tau in place of pi restores it.
N.B. I made a previous comment along these lines, but put it in the wrong place. If you're still a "pi is wrong" skeptic, considering the improvement in radian angle measure may yet convince you:
The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc.
In these terms, Euler's formula would be recast as
e^(iτ) = 1
That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
I've thought pi is horrible... and adopted using a loopy pi as 2pi. I loop the first vertical line over the horizontal, right, and then make it the horizontal, and so on. If that makes any sense.
I'm thinking of making a LaTeX package containing it... (and also some other personal conventions)
Also, to strengthen your point, 2pi makes the radian system a lot more natural.
I made up the usage of τ myself, partially because of its typographic similarity to π, partially because it leads naturally to the usage "τ = turn". It's too bad that π has two legs while τ has only one; it would be poetic if π were 2τ, but it wasn't to be.
N.B. I have a secret master plan to spread the use of τ, but this comment is too small to contain it. ;-)
All of those (except the circle) have the 1/2 because they're integrals of something linear. While it's true that area and integral are closely related (the latter being a special case of the former), a circle is clearly not linear.
johnaspden has it right. To put it more explicitly, we can calculate the area of a circle by integrating the differential element of area dA for an infinitesimal annulus from 0 to r. Now, dA is simply the arclength (circumference C) times the thickness dr; since the circumference scales linearly with radius, this leads to the integral of a linear function as follows:
This is actually my favorite example of why pi is wrong—it's the "exception" that proves the rule. To see why, set τ = C/r = 2 pi, and then consider the following chart of common quadratic forms:
We see that, far from causing "different complications", using the right circle constant brings the area of a circle into a more natural form. The 1/2 in the formula for circular area is actually a missing factor; using tau in place of pi restores it.N.B. I made a previous comment along these lines, but put it in the wrong place. If you're still a "pi is wrong" skeptic, considering the improvement in radian angle measure may yet convince you:
The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc.
In these terms, Euler's formula would be recast as
e^(i τ) = 1
That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.