Let us consider a circle. A useful definition of a circle is the set of all points which lie the same distance from a single point. You can easily make a circle by using a pointy sharp thing with a fixed length of string attached to it, and with some sort of device that leaves a mark attached to the end of the string. You also need a flat surface. You can draw circles on non-flat surfaces, like footballs or saddles, but that way lies a special kind of madness called non euclidean geometry, which I am not touching with a bargepole. You stick the pointy sharp bit into the surface you wish to draw a circle on, pull the string tight, and place the marking device on the surface. What you have marked is a point which is the string's length away from the pointy bit. Now if you lift the marking device up and place it down (with the string still held taught) ANYWHERE else, you will mark another point the string's length away from the pointy bit. If you keep doing this randomly, you will eventually see the outline of a circle start to form - defined by all the individual points.
Of course, that is not the sensible way to draw a circle. Instead of lifting the marking device up every time, you just leave it touching the surface and, again keeping the string tight, you move it in either direction. I say either direction because you will find that you are constrained to only move two ways, clockwise or counter clockwise. Once you have moved back to the point you started from you have drawn a circle. Well done. Now, we can say five things about the circle you have marked. We can of course say what colour it is, but that is irrelevant for geometrical purposes. Secondly we can say how thick the line is that marks out the circle. This will depend on your choice of marking device. A felt tip pen or highlighter will leave a thicker line than a biro, and a crayon or a piece of chalk will leave a thicker line that all of the foregoing.
In pure geometry though, lines do not have thickness, and points do not have an area. A line, including the outline of the circle we drew, only extends in one direction. This makes sense, because of our definition., The circle is the set of all points EXACTLY the same distance from another point, not sort of the same distance depending on how big your marking device is. So the thickness of the line marking the circle, as with its colour, is irrelevant to us.
Thirdly, we can say how big the area is inside the circle. This is going to be measured on a two dimensional surface, so the answer will be some number of whatever units you like to the power of two. By that I mean the units are squared, not the number of them. So if your chosen measure of length is the flangit, then your chosen measurement of area will be square flangits. A square flangit is just a square whose length is exactly a flangit. Squaring a number is the same as raising it to the power of two. The area in the circle is therefor some number of square flangits.
Fourthly, we could talk about the length of the line we have drawn on the surface. If we took it in our minds eye and straightened it out, and measured it, how long would it be? Fifthly and finally we can talk about the length of the bit of string between the pointy sharp bit and the marking device.
As it turns out, the area in the circle and the length of the line you draw are related ONLY to the length of that bit of string. If that seems remarkable, remember that it doesn't matter where on your surface you poke your pointy sharp thing your circle will still look the same. The only determining factor of the size of the circle you create is the length of string you allow between the pointy sharp bit and the marking device.
So what does all this have to do with \(\pi\)? Well, if you take your length of string, and multiply it by \(2\pi\) you get the length of the line you draw. So \(\pi\) is the RATIO between the length of the string and the length of the line around the outside of the circle you draw with that string. It matters not a jot how long your bit of string is, the circle that results will ALWAYS have a line that is \(2\pi\) times the length of that string long around its outside.
So why do we say \(2\pi\) and not just \(\pi\) for this ratio? Good question. It turns out that it is a lot more useful to work with \(2\pi\) because \(\pi\) itself turns up in far more places on its own. If we went with \(\pi\) then we would keep having to half it. For example, the area of the circle that we can describe? It is the length of the string times itself (this makes the units squares remember) multiplied by \(\pi\). If we defined \(\pi\) as the direct ratio between the length of the string and the line, that number used to find the area would have to be half of \(\pi\). Which looks messy.
The line which forms the circle is called the circumference of the circle. Another word for something that runs around the outside of something else is the perimeter. The word for perimeter in ancient greek started with a letter in the ancient greek alphabet. Can you guess which letter? The length of the bit of string is called the radius of the circle. The point we made with the pointy sharp thing is the centre of the circle.
There is a line which is double the length of the radius, and is described as the straight line from one point on the circle, through the centre, to another point on the circle (which will inevitably be exactly opposite the starting point). This line is the diameter of the circle. Because this line is the radius multiplied by two, and because (as we have seen) it does not matter which order you multiply things in, you can get rid of the two from the \(2\pi\) when you talk about the ratio between the diameter and the circumference. The circumference is therefor just \(\pi\) times the diameter. This is because the radius times two times \(\pi\) is the same as two times the radius times \(\pi\), and two times the radius is the diameter.
What does this all look like? Glad you asked. Have a gander at this snazzy diagram:
I don't like the diameter though, because it has nothing to do with the construction of the circle. The diameter comes about once a circle has been made, while the radius is used in making the circle in the first place. So I prefer to think of \(\pi\) as it relates to the radius not the diameter.
The number itself is a bit more than three. It is not a whole number you can count on your fingers, and neither is it a fraction (although \(\tfrac{22}{7}\) comes pretty close. If you were to write out \(\pi\) as a decimal number as three point something something something, you would be writing "somethings" for ever, because there are an infinite amount of them. So instead of worrying about all those "somethings" we just write \(\pi\). This is a fantastic animation of the circumference "unrolling" to show it is \(pi\) times the diameter:
That animation was created (and GPL licensed) by wikipedia user John Reid.
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