How is Pi calculated?

[greenpeace]

Before I get to the point, a mod basically told me that Science/Tech includes math so this is in the right forum (they should probably make it clearer though).
So as I said in the title, how is Pi calculated, and whats the proof that the calculation actually gives the ratio of the circumference of a circle to its diameter?

 

[Aramazd]

Wikipedia is your friend:
http://en.wikipedia.org/wiki/Computing_π
http://en.wikipedia.org/wiki/Computing_π

 

[greenpeace]

Wow, I can't believe i didn't think of that, thanks for pointing out the obvious for me.

 

[peter grimes]

How about a non-wikipedia source?


I remember reading somewhere (ages ago) that one of the earliest ways it was computed involved a wheel and a flat surface, like a beach or field:

A line is marked on the ground as the starting point (so you know where you began)
A tick mark is made on the face of the wheel, near the edge, where the wheel sits on the starting line (so you know where the circumference starts)
Roll the wheel along the beach, making a mark in the sand where the tick mark touches the ground.
Lay the wheel on the starting line, with an edge touching the starting line, and the greatest diameter of the wheel laying squarely over the track of the wheel left from the rolling step. Make a mark on the track where the opposite side of the wheel's edge touches the track.
move the wheel to the new mark, and repeat.

Keep laying the wheel along the track until the edge of the wheel lines up with one of your rotational marks. Know where they're going to line up?

Obviously, the longer your track, the greater the resolution of your fraction.

 

[Aramazd]

How about a non-wikipedia source?

It's accurate, so I don't see the problem

Spoiler :

I remember reading somewhere (ages ago) that one of the earliest ways it was computed involved a wheel and a flat surface, like a beach or field:

A line is marked on the ground as the starting point (so you know where you began)
A tick mark is made on the face of the wheel, near the edge, where the wheel sits on the starting line (so you know where the circumference starts)
Roll the wheel along the beach, making a mark in the sand where the tick mark touches the ground.
Lay the wheel on the starting line, with an edge touching the starting line, and the greatest diameter of the wheel laying squarely over the track of the wheel left from the rolling step. Make a mark on the track where the opposite side of the wheel's edge touches the track.
move the wheel to the new mark, and repeat.

Keep laying the wheel along the track until the edge of the wheel lines up with one of your rotational marks. Know where they're going to line up?

Obviously, the longer your track, the greater the resolution of your fraction.

There is an easier way that only requires paper, if you know the radius. A circle is x^2+y^2= r^2. In terms of y, y=(r^2-x^2)^(1/2). Integrate this and divide by the radius squared.

 

[Chairman Meow]

Dang kids making me dig out all my old pi links, just to find most of the links are dead. Now I have to dig up new links.

You can find a good summary of various methods of calculating Pi here. In addition, one method that is particularly interesting is the BBP Formula, which allows one to calculate digits of Pi in base 16 without needing to know the preceding digits. In addition, there are various sums that are used to calculate Pi.

 

[peter grimes]

There is an easier way that only requires paper, if you know the radius. A circle is x^2+y^2= r^2. In terms of y, y=(r^2-x^2)^(1/2). Integrate this and divide by the radius squared.

The problem with that method is you need some relatively advanced technology to compute it.

The method I outlined relies on a mechanical / geometric solution. The Greeks and Egyptians didn't have calculus, but they were masters of ratios