I understand that it IS pi, but I don’t understand what “ratio of the circumference of any circle to the diameter” means.
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Let’s say you have a tape measure. If you wrap that tape measure around a circle and measure it, that’s the circumference. If you take the tape measure and measure the circle across, from one end to the other, that’s the diameter. No matter how big the circle is, if you divide the circumference by the diameter, you always get pi.
Circumference and diameter are two measures of a circle. Circumference is the length around the outside, and diameter is the “width” if you will, from one part on the edge, through the center, to the opposite side.
For a circle with a diameter of 1 unit, the circumference will be pi units. That is, if you take a string with the length of the diameter and wrap it around the circle, it will take a total of 3.14… lengths to make it all the way around. *edited for better phrasing
“Ratio of the circumference of any circle to the diameter”
Means that you could take any circle, doesn’t matter how big or small, the circumference will be 3.1415… times as big as the diameter.
It means that this is an inherent property of all circles.
Its easiest to understand by [seeing it visually here](https://upload.wikimedia.org/wikipedia/commons/4/4a/Pi-unrolled_slow.gif)
Think of it not as a circle to start but as a single straight line. What happens when you curl that straight line up into a circle?
Any circle will be the same. Pi. When you uncoil a circle, you always get the ratio of circumference to diameter is 3.14…
All circles are similar to each other. That is, they all have the same shape regardless of how we scale them up or down. Squares have the same property. If you measure the width of a square, you will find that it is always 1/4 of the perimeter.
Imagine the circle is the top of a barrel. You can take a measuring tape and wrap it around the top to measure the outside – the circumference. You could also use the measuring tape to find the distance across the circular top – the diameter.
If you divide the number you measured for the circumference by the number you measured for the diameter (that’s a ratio), you get Pi (3.141592…). It doesn’t matter how big the circle is, the circumference is always Pi times as big as the diameter.
That’s how we define Pi. When people recognized that the circumference divided by the diameter of the circle never changed (it is “a constant”) they figured that was a useful number to know and used the greek letter as a nickname for it.
Measure the circumference of a circle. Then measure its diameter.
Divide the circumference by the diameter, you’ll get pi.
That’s it. It just means the circumference divided by the diameter.
Let’s say instead of math problems you are dealing with something real. Maybe you have a swimming pool that is perfectly round. You want to place a rope along the edge of the swimming pool all the way around. Why are we doing this? I dunno. It’s more interesting than math problems. You don’t know how long it is around because the only thing you have to measure is straight and it reaches across the pool from one end to the other, right over the center of the pool. The pool is 10 feet across. That’s a diameter. Your pool is 10 feet across.
It seems to make sense that since we know how “big” the circle is, we can also know how far around it is. There is a way to calculate it. That number is pi.
So how much rope do we need? Well, we figure we’ll need at least 10 feet since it’s longer around the whole thing than it is across. So we lay out 10 feet of rope along the edge of the pool. It goes just about a third of the way around. If we envision it like a piece of pie (unrelated word, but the ramification are uncanny), the rope the length of the diameter of the pool, takes up about not quite one third of the whole pie, like a pie cut into pieces for three people (about 120 degrees each slice).
But that’s not big enough to go all the way around, how about another piece of 10 foot rope – that goes around almost 2/3 the way around the pool. Almost. We’re getting there. So let’s get another piece of 10 foot rope and add it, this look like it might do the trick. Except it doesn’t. We put out 3 pieces of 10 foot rope around the pool of 10 feet diameter across, and we have a little bit left. It appears to be about a foot left. So now we cut off a foot of another piece of 10 ft rope and add that. So we have 3.1 pieces of 10 foot rope going around the pool. Almost, there’s still a small gap. We measure it and we figure it is 0.04 the size of the 10 foot piece of rope, so we cut that off and add it. But wait, there’s still a little tiny gap left. For practical purposes were probably done and we can round off and call it a day. For the math problem though, we are not.
The problem isn’t really that pi is “irrational” but rather than we are trying to represent pi in our system which is based on 10. Trying to jam pi into pieces of 10 doesn’t work, there’s always a little tiny bit leftover.
But that’s the idea – the way around the circle is a little bit more than 3 times across the circle.
Think of a square. No matter how large or how small you make it, all 4 sides are the same size, so the perimeter of the square is 4 times the width of the square. In other words, the ratio of the perimeter to the side of the square is 4.
Now think of a circle inside that square such that the edges of the circle touch the sides of the square. In other words, the diameter of the circle is equal to the side of the square. The circumference of a circle is just the perimeter of the circle. By looking at the picture of a circle inside the square it’s immediately obvious that the perimeter of the circle is smaller than the perimeter of the square, therefore the perimeter of the circle (the circumference) has to be less than 4 times the side of the square (the diameter of the circle). How much less? That is the question. Well it turns out that it’s about 3.14 (pi) times. In other words, the ratio of the perimeter (aka circumference) to the side (aka diameter) of the circle is pi.
By the way, to add to it: “ratio” in math means a relationship between two numbers. It works out as though you’re setting them up as a fraction:
– The ratio of 6 to 2 is 3. (6 is 3 times bigger than 2. Or you could just say 6/2 = 3.)
– The ratio of an 1,000 foot building to a 500 foot building is 2. (1,000 is 2 times bigger than 500. Or you could just say 1000/500 = 2).
– The ratio of the circumference of a circle to its diameter is about 3. Actually it’s exactly 3.141519…. (Circumference/diameter = pi)
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