Why is pi defined as C/d instead of C/r?

[StatusX]

It seems a lot more natural to define pi by the ratio of the circumference of a circle to the radius, rather than to the diameter. The diameter is hardly ever used in math (at least in my experience), and it seems like most formulas involving pi involve the combination 2pi. Is it a historical accident that we use pi=C/d, or is there a good reason for it?

 

[sebas531]

.........yeah, I think there's a really good reason for it, if they use radius without multiplying times two then that would not give the true circumference which is what we are relating pi to. Thus not giving us a true value of pie. Although one cannot achieve an exact value of pie. If you really want to see radius there then you can always write pi=C/(2r). One can also define pi as pi=A/(r^2) excuse me if this was not what you were asking for.

 

[Jack Bauer]

expressing the relation in terms of the radius doesn't say anything new, since the relation of d=2r is already known. Why not keep it as simple as possible?

 

[Data]

I'd just consider it a historical accident . And besides, I don't like fractions, so remembering \frac{\pi}{2}r^2 for the area of a disc would make me sad!

Not to mention the greatest tragedy of all, the destruction of the beauty of Euler's relation e^{i \pi} + 1 = 0.

e^{\frac{i \pi}{2}} + 1 =0 just doesn't look the same!

 

[robert Ihnot]

YOu may have a good point in that. I quote about Euclid: He was able to show that the perimeter of the polygon was proportional to the radius (which is half of the diameter), regardless of its size. He then increased the number of sides of the polygon, realizing that as he increased them, the perimeter of the polygon got closer and closer to that of the circle. Therefore, he was able to prove that the perimeter of the circle, or circumference, is proportional to the radius and also to the diameter. http://www.arcytech.org/java/pi/facts.html

 

[HallsofIvy]

Suppose you have a column or tree trunk to deal with. Which is easier and more natural to measure, the diameter or radius?

 

[Repetit]

Just a question about the destruction of the euler equation that Data talked about. If pi was defined as the ratio between circumference and radius wouldn't that mean something for cos and sin also? So that cos(\pi)= 1 and sin(\pi)=0, and the euler formula therefore is preserved? Or am I completely wrong?

 

[Office_Shredder]

Just a question about the destruction of the euler equation that Data talked about. If pi was defined as the ratio between circumference and radius wouldn't that mean something for cos and sin also? So that cos(\pi)= 1 and sin(\pi)=0, and the euler formula therefore is preserved? Or am I completely wrong?

It would be e^{i \pi} - 1 = 0

 

[Data]

yes, i suppose e^{i\pi} = 1 is almost as good :) you miss out on the additive identity though!