eli5 Why is the circumference of a circle divided by its diameter always (for the majority) equal to pi?
Because that is how we’ve defined pi.
Circles are always consistently shaped, and making them larger/smaller doesn’t change the ratio of Circumference:Diameter.
It’s the same reason a square’s perimeter is always 4x its length. It has 4 equal sides, each with the same length, so the total of the sides is 4x its length.
A circle is a consistent shape so its proportions are always the same. A circle has smoothly curved sides instead of straight sides like a square, so the math to calculate that ratio isn’t as simple. But that ratio is pi, and every circle is *exactly* the same as any other circle, just bigger or smaller, so the comparison between its circumference and diameter is always the same.
It’s not equal to pi “for the majority,” it’s flat-out equal to pi.
If you were to take a perfect circle–*any* perfect circle, of any size–and cut it, and then straighten it out into a line, you will find that the length of that line is *always* exactly equal to pi times the diameter of the circle.
If you want to get more technical about it, you can look to calculus. When you have a function, you can take a “derivative” of that function to tell you how it changes with respect to some other variable.
In this case, the function is f(**d**) = π**d**, where d is the diameter of the circle. This gives you the circumference of a circle. If you take the derivative of that function with respect to the diameter (that is, if you want to see how the circumference changes as the diameter does), you get this: d/d**d** πd = π, or f'(d) = π. (“**d**” for diameter is bolded here for ease of reading. The function could also be written f(x) = πx, and d/dx πx = π).
…because that’s what pi is?
Pi is not a special numbers, it is what it is. Pi could have equaled just 3.12345 and ended at just 5 decimals and it would be just as prevalent in math. The fact that it goes on forever without repeating is just additional, there are a ton of other numbers that do it as well (such as *e*). We use the symbol because we can’t right it out in a formula (we probably would also use it even if it equalled 3.12345 because that too is pretty long to do multiple times in a workout).
Because that’s how we defined pi. It’s defined as the ratio of circumference to diameter and since you can’t change one without changing the other proportionately, that ratio never changes.
There is no circle in which the ratio of circumference to diameter is not pi. All circles are exactly the same, magnitude does not change proportions. I’m not sure why you added “for the majority”.