Why did the first person even ask what it was?
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We had rules for a square, triangle rectangle etc to ge the area, they would have noticed the relationship between the square of the radius and the circumference, for smaller numbers it is roughly 3 times the quantity, they would have known that it is not exact though if they used something like a rope ruler to measure out a circle with a circumference of small length.
I imagine one of the earliest uses was with wheels. If I know how big the spokes (diameters) of my wheels are, then I need to find out how much to material I need to curve for the rim (circumference) to make a functioning wheel. This would be very important for things like chariots, which were used as far back as 3000 BCE. The better your wheels are the smoother the ride in the chariot, and the faster you can go with the same horses – all important things for the luxury cars of the age.
because it determines whether the circle is big or small depending on the length of the diameter. Remember that the diameter is not only just a line, it is a line that defines the length from one edge to the other of a circle, so if that distance is smaller, the circle becomes smaller. Lemme explain with an example, if you have a pie and you cut it into slices from the middle and then you take a slice, the length is the radius, all the slices have the same length, so if the slice gets smaller and all slices are equal, then the pie itself reduces in size. The same is true if you grab two slices which make up the diameter and you reduce the length of them, then the pie will inevitably reduce in size.
Idk who asked first, but the ratio was probably used without even knowing the formula in the first place through a rudimentary form of a compass. The Egyptians for instance, used sticks and tense rope to make a compass and thus make circles, the length of the rope functioning as a radius.
It’s not important. But if you know the diameter you can calculate the circumference and vice versa. This can be very useful in many industries and professions.
I guess you are asking this historically, right? Like, how did we even find out? Scholars in ancient times did study geometrical shapes and they found out. Why did they study geometry? Curiosity… humans are big curious monkeys and we want to understand (and control) the world we live in.
>Why is the ratio of a circle’s circumference to its diameter important?
It’s not important at all. What is important is the ratio of a circle’s circumference to its radius. After all, a circle is defined by its radius. The fact that there is a ratio to the diameter is merely a side effect.
complex analysis is based on 2 dimensional coordinate. it can be represented using radius and angle. so pi comes everywhere there
People building things would have just noticed that the ratio was always the same.
You need to build something circular. To ensure its accuracy (especially if you need to build more than one), you make a jig or pattern. To get started, it’s very easy to sweep out a circle with a nail, a string, and piece of chalk. You need to figure out how much material you need, so you take some measurements of your circle and find out how wide it is and at least a “close enough” estimate of the circumference.
Once you’ve done this on a few different sized circles, you’re going to notice that the circumference is always about three times the diameter. And it’s going to be worth it for you to notice this because now all you have to do is measure the diameter and you know everything about the circle without fiddling with whatever more tedious method you were using to measure the circumference.
Now, you’re a tradesman so you figure it’s 3 or a little more and that’s all you need to know. Then someone like Archimedes rolls around. It’s known there’s this ratio, and no one knows exactly what it is, but everyone is pretty sure it’s always exactly the same. So Archimedes (or someone like him) proves that it is always the same and develops a method for calculating it more accurately. He comes up with 22/7 and that’s close enough for any practical purpose that doesn’t leave the Earth’s atmosphere even today.
Pi is important because it shows up everywhere. This is because trigonometry is based on the unit circle. And if you build something from a circle, pi is simply going to pop out of it at some point. Plus, anything that vibrates or oscillates–anything with a frequency or a wave–can be expressed with sines and cosines. Pi is just baked in to the way we built higher mathematics.
Because it allows you to calculate one from the other. This is useful where it’s not easy to calculate one. An example is a circular structure. If you need to know how many bricks you need, it’s easier to measure the diameter than measuring the perimeter.
Like other people have said, it’s extremely important for basically any calculation involving a circle.
This also means that it shows up ***everywhere*** in seemingly unrelated math. If something can potentially be described as any property of a circle, you can get pi out of it.
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