Why is pi = circumference / diameter ?

[Juwane]

Why do we take pi as the ratio of the circumference to the diameter, and not diameter to the circumference? Is it because circumference is always bigger than the diameter, so that it will be easy to work with the ratio? Or is it something fixed by those who discovered it and we can't change it?

 

[farful]

You also should realize that this is the ratio of the circle's area to the square of its radius. That is, a unit circle has an area of pi.

 

[elibj123]

It's just a definition...

 

[Juwane]

What if we turn it upside down; that is, diameter over circumference. This is still a ratio, isn't it?

 

[rasmhop]

What if we turn it upside down; that is, diameter over circumference. This is still a ratio, isn't it?

That would be perfectly acceptable as well. pi is just a number someone defined (probably someone from ancient Greece), but you can define lots of other numbers related to it if you want. At some point we defined pi and since then we haven't found a major reason why we would rather have diamter/circumference or anything else related to it. In some areas of math we more often work with \sqrt{\pi}, 2\pi or 1/\pi than \pi, but we don't re-define pi or introduce a new symbol since it wouldn't be worth the trouble.

 

[hamster143]

Why do we take pi as the ratio of the circumference to the diameter, and not diameter to the circumference? Is it because circumference is always bigger than the diameter, so that it will be easy to work with the ratio? Or is it something fixed by those who discovered it and we can't change it?

It's a 2500 year old convention.

 

[Borek]

You may as well ask why pi and not rho

 

[wofsy]

Why do we take pi as the ratio of the circumference to the diameter, and not diameter to the circumference? Is it because circumference is always bigger than the diameter, so that it will be easy to work with the ratio? Or is it something fixed by those who discovered it and we can't change it?

In ancient times geometers probably thought of the perimeter of a figure a derived or constructed from a unit length and so expressed the perimeter as a multiple of one. A circle is well approximated as a perimeter of a many sided regular polygon.

From this point of view the unit length would be the radius of the circle so I would guess that the original formula was circumference/2xradius.

 

[farful]

You may as well ask why pi and not rho

Because it's the first letter in the word "perimeter".

 

[Borek]

That still puts us in the world of conventions. What if pi was defined earlier by Egyptians? Or later by Muslims? It would be as obvious that we use some different symbol as it is that we use pi now.

 

[zgozvrm]

I don't know, why do we call a chair a "chair" and not a "kanildor?"

 

[Borek]

Convention