Someone measured a circle’s circumference and they measured its diameter. They noticed that the two numbers always had the same ratio no matter how big the circle was, so they gave that specific ratio a name so that they could actually talk about it without every conversation being about “That ratio between a circle’s circumference and it’s diameter” which is a mouthful.
Fun fact, for a while, it was believed that pi was equal to 22/7 until we got more accurate tools and discovered that that’s slightly off.
The earliest places it would’ve shown up would be construction.
Let’s say you want to build a circular temple. You need to figure out how much building material you need. You know how far across the area it is you’re building, so with that diameter, how far around is it?
You could just place a string around a 1m diameter circle. Measuring the string you get approx 3,15 meters.
Doing the same with a 10 m radius circle gives you 31,4 meters. Now you know it’s definitely a fixed ratio for all circles.
You also now know that reducing a square by cutting corners, maintaing a perimeter of 4, towards the shape of the circle, can’t be a good proof for a circle’s circumference.
Even very early (like over 4000 years ago) people knew that the relation of the diameter and circumference was close to 3 but not a whole number.
One of the eisiest way to figure out pi is to put a circle that is just larger than a square and just smaller than a hexagon. Then with some simple geometry you can approximate pi between the “pi for a square” and “pi for a hexagon” (which you can calculate exactly).
Before its exact value was known, it was used by mathematicians to be the ratio of the circumference of a circle to its diameter.
It was first calculated by a Greek mathematician Archimedes, who started with a unit square (i.e. all sides are of length 1), and continuously added more sides, making it a hexagon, octagon and so on. The perimeter of the square is 4, meaning pi is less than 4. It gets a bit tricky, but with trigonometry rules known at the time it was possible to calculate this to quite a bit of sides (I think he went up to 96). The more sides that were added, the more the perimeter decreased while the width (what will become diamater) remained constant. This works because a circle can be thought of as a polygon with infinite sides, so the more sides of a polygon you have, the closer it gets to a circle. Eventually, the perimeter would level out at what we know as pi.
Interestingly, at that time, decimal numbers weren’t being used yet, so while a value for pi was known, it was only known as a ratio. The number 3.14…. was calculated much later and to many more digits than would have been known at the time of Archimedes
You can estimate pi by estimating the circle as a polygon. A hexagon is closer to a circle than a pentagon is, etc. You can exactly calculate the length around it, if its diameter is 1. Then add more sides. You can add as many as you’re willing to calculate, and you’ll get closer and closer to the circumference of the circle (which will be pi, if the diameter is 1).
AFAIK people had done it by hand out to like 200-gons by the middle ages (during the Islamic Golden Age, when they also invented algebra and stuff).
In the beginning you have someone with an obsessive interest in measuring circles. Then you happen to notice that the radius is always seems to be about 3*r^2
Then once you’ve noticed that pattern you see if it’s true everywhere.
“And one of these facts is this: By 2,000 B.C., men had grasped the significance of the constant that is today denoted by π, and that they had found a rough approximation of its value.
How had they arrived at this point? To answer this question, we must return into the stone age and beyond, and into the realm of speculation.
Long before the invention of the wheel, man must have learned to identify the peculiarly regular shape of the circle. He saw it in the pupils of his fellow men and fellow animals; he saw it bounding the disks of the Moon and Sun; he saw it, or something near it, in some flowers; and perhaps he was pleased by its infinite symmetry as he drew its shape in the sand with a stick.”
Dont know if we know, but all it takes is for someone to notice that when their cart wheel rolls around once, their cart has moved three times, and a bit, as long.
They didn’t decide the value of pi, they calculated it
first they noticed that the circumference of a circle was always around 3 times longer than it’s diameter
so they started to try get better estimates on this new constant using techniques involving polygons and got to around two digits (3.14)
Then math formulas were discovered for calculating pi and this was a big speedup, they got pi to a few hundred digits
After computers were invented, their estimations of pi got ALOT more quick and it went from thousands to trillions of digits
If you take the circumference of a circle and lay it out flat like a line, the line will be slightly more than 3x the diameter of the same size circle. If you do the math out properly, that slightly more than 3x diameter is exactly π.
The next time you buy a pizza, measure de diameter and divide it by 2, cut it in narrow slices, 20 for example. Grab them and put them side by side, alternating one with the crust towards you and the next with the crust opposite, measure the crust length on one side and multiply it by two (to get the length of both sides). It will be 3.14 times the radius. If your pizza was 15 cm radius, you’ll have ~47 diameter/length.
By arranging the slices that way you’re measuring the circumference in an easier, linear, way.
The more slices you have, the more perfect the circle of the pizza is made, the more accurate your result will be. They needed a way to measure the surface area and the volume of round objects and for that you need to know the ratio of the radius to the circumference which is called pi. It will always be 3.14 radiuses, no matter the size of your pizza. Like on a square, all the sides are the same length to each other, no matter how big or small the square is.
You can do this yourself! Get some string and a ruler and go around measuring the outside (circumference) of different circles you find, and then measure across the middle (diameter). If you divide the circumference by the diameter, you’ll find that they’re all about 3 and a bit.
Measuring more precisely will get you closer to π.
Ancient Greek philosophers were incredibly clever mathematicians. They asked really smart questions like “what is the ratio of a circle’s circumference to its diameter?”
Well they didn’t have fancy math tools like calculus or computers but they didn’t have a compass, a straight edge, and a lot of time to think. Since they couldn’t directly measure a curve like we can now, they took to estimating it. They figured out very quickly that all circles seem to have the same ratio. They simply called this ratio “pi”.
The ancient Greeks came up with a plan. If all you have is a compass and straight edge all you can draw are circles and lines. So you start building easy shapes like triangles and squares and hexagons oh my! You can “squeeze” a circle between 2 polygons, one larger and one smaller, so that their easy to calculate perimeters and diameters give you upper and lower bounds on the circle’s circumference. If you set the diameter equal to 1 then the circumference must itself be pi. Using this method they got 3.16 which ain’t too bad for 2500 years ago.
The problem is that we don’t even know if pi could be written as a fraction. It wasn’t until the development of what we might call “modern” mathematics did we prove that pi is irrational (can not be written as a fraction) and then awhile later we proved that it must be transcendental (can not be the solution to a special family of functions called polynomials).
So we can’t write it as a fraction and it can’t be found from standard equations you might learn in school so what can we do? Well we can do what the ancient Greeks did but better because we have supercomputers now. We have come up with incredibly fancy ways of calculating estimations of pi to trillions of decimals and there’s still more we don’t know! For example, we don’t know if pi is a normal number. This would mean that if you counted how many times each numeral appears (how many 1s you see, how many 2s, 3s, 4s, etc.) you should get them all appearing roughly the same number of times.
The start plugging random numbers into the place of pi if the answer you get is too big than make that number smaller if it’s too small than make the number bigger. So…
3 too small
4 too big
3.5 too big
3.2 too big
3.1 too small
3.15 too big
3.14 too small
And continue doing that forever until you get as many digits as you want.
What I think is funny, when the ancient geometry/math guys did this, how pissed off were they that not only did it not fit a nice round number, but it didn’t fit any number?
Archimedes of Syracuse (that’s a Greek fella from over 2250 years ago) was the first jack to write it down that we know of.
He drew a circle and broke it down into straight lines, like drawing a pentagon, hexagon, etc around a circle.
Turns out, the radius averages out to coincide with the circumference. And the more angles it takes, (like going from a hexagon to an octagon) the more accurate the ratio for ‘Pi’.
The ancient Greeks were pretty good at writing shit down, though. So, it’s possible that this “ratio” had been discovered previously by mathematicians from other areas and or eras.
According to my calculus professor years ago, if we were to find the final number of pi, we would be able to put a round roof on a square building with zero overhang from either object.
I can’t remember all the details, but one of thr most impactful college courses i took started with some lectures on how ancient civilizations estimated Pi. All around the same time, you got estimates of 3.16, 22/7, etc. By various methods, they were all somewhat close, one being as close as 3.1416 in ancient India if I remember correctly.
Most notably, an ancient East Asian culture estimated it by comparing the longest chord of a regular polygon with the measure of its perimeter. I think they were able to get up to a 17 sided polygon before being unable to calculate further, and correctly had the first 7 digits of pi! It was so crazy because each culture was close, but this one was so incredibly accurate
They calculated the sum of the sides of a square with a diagonal length of 2 (each corner is 1 from the center point in all of these), then a pentagon, then a hexagon, then a septagon, an octogon, and on and on until they could see a pattern. As the number of sides increased, the closer the answers got. They called the answer that they would finally reach infinite, a circle, sides as pi. Later, some very smart people with a ton of time on their hands and the invention of calculus they were able to figure out the actual value.
How is this taught in every single school in the world? You have enough intelligence to access the internet and type the question on reddit, but never learned how circles work? What?
1 Kings 7:23 23
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it. 24 Below the rim, gourds encircled it—ten to a cubit. The gourds were cast in two rows in one piece with the Sea.
2 Chronicles 4:2
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits[b] high. It took a line of thirty cubits[c] to measure around it.
Comments
They divided a circles circumference by its diameter and realized the answer was always the same.
Someone measured a circle’s circumference and they measured its diameter. They noticed that the two numbers always had the same ratio no matter how big the circle was, so they gave that specific ratio a name so that they could actually talk about it without every conversation being about “That ratio between a circle’s circumference and it’s diameter” which is a mouthful.
Fun fact, for a while, it was believed that pi was equal to 22/7 until we got more accurate tools and discovered that that’s slightly off.
Measure the circumference of a circle and divide that by the diameter of the same circle. No matter the size of the circle, you get pi as your answer.
The earliest places it would’ve shown up would be construction.
Let’s say you want to build a circular temple. You need to figure out how much building material you need. You know how far across the area it is you’re building, so with that diameter, how far around is it?
I’m sure it stared with a wheel of cheese and a LOT of wine/mead/liquor.
“Man, I stg, no matter what wheel of cheese I measure, the outside is always around 3.1-3.2 times longer than the wheel is wide”.
Then, as more precise measurements came out, it got whittled down to 3.14. Then 3.145. Etc as nasuem.
I’m likely wrong on so many levels. But this is how I picture it happening.
[removed]
You could just place a string around a 1m diameter circle. Measuring the string you get approx 3,15 meters.
Doing the same with a 10 m radius circle gives you 31,4 meters. Now you know it’s definitely a fixed ratio for all circles.
You also now know that reducing a square by cutting corners, maintaing a perimeter of 4, towards the shape of the circle, can’t be a good proof for a circle’s circumference.
Pi is equal to the number (divided by 4) of people answering without checking the comments first.
Even very early (like over 4000 years ago) people knew that the relation of the diameter and circumference was close to 3 but not a whole number.
One of the eisiest way to figure out pi is to put a circle that is just larger than a square and just smaller than a hexagon. Then with some simple geometry you can approximate pi between the “pi for a square” and “pi for a hexagon” (which you can calculate exactly).
Before its exact value was known, it was used by mathematicians to be the ratio of the circumference of a circle to its diameter.
It was first calculated by a Greek mathematician Archimedes, who started with a unit square (i.e. all sides are of length 1), and continuously added more sides, making it a hexagon, octagon and so on. The perimeter of the square is 4, meaning pi is less than 4. It gets a bit tricky, but with trigonometry rules known at the time it was possible to calculate this to quite a bit of sides (I think he went up to 96). The more sides that were added, the more the perimeter decreased while the width (what will become diamater) remained constant. This works because a circle can be thought of as a polygon with infinite sides, so the more sides of a polygon you have, the closer it gets to a circle. Eventually, the perimeter would level out at what we know as pi.
Interestingly, at that time, decimal numbers weren’t being used yet, so while a value for pi was known, it was only known as a ratio. The number 3.14…. was calculated much later and to many more digits than would have been known at the time of Archimedes
You can estimate pi by estimating the circle as a polygon. A hexagon is closer to a circle than a pentagon is, etc. You can exactly calculate the length around it, if its diameter is 1. Then add more sides. You can add as many as you’re willing to calculate, and you’ll get closer and closer to the circumference of the circle (which will be pi, if the diameter is 1).
AFAIK people had done it by hand out to like 200-gons by the middle ages (during the Islamic Golden Age, when they also invented algebra and stuff).
In the beginning you have someone with an obsessive interest in measuring circles. Then you happen to notice that the radius is always seems to be about 3*r^2
Then once you’ve noticed that pattern you see if it’s true everywhere.
I recommend A History of Pi, by Petr Beckmann.
Great book.
Edit:
Excerpt:
“And one of these facts is this: By 2,000 B.C., men had grasped the significance of the constant that is today denoted by π, and that they had found a rough approximation of its value.
How had they arrived at this point? To answer this question, we must return into the stone age and beyond, and into the realm of speculation.
Long before the invention of the wheel, man must have learned to identify the peculiarly regular shape of the circle. He saw it in the pupils of his fellow men and fellow animals; he saw it bounding the disks of the Moon and Sun; he saw it, or something near it, in some flowers; and perhaps he was pleased by its infinite symmetry as he drew its shape in the sand with a stick.”
Excerpt From
A History of Pi
Petr Beckmann
https://books.apple.com/us/book/a-history-of-pi/id949739510
This material may be protected by copyright.
Dont know if we know, but all it takes is for someone to notice that when their cart wheel rolls around once, their cart has moved three times, and a bit, as long.
They didn’t decide the value of pi, they calculated it
first they noticed that the circumference of a circle was always around 3 times longer than it’s diameter
so they started to try get better estimates on this new constant using techniques involving polygons and got to around two digits (3.14)
Then math formulas were discovered for calculating pi and this was a big speedup, they got pi to a few hundred digits
After computers were invented, their estimations of pi got ALOT more quick and it went from thousands to trillions of digits
More generally, was mathematics invented or discovered?
If you take the circumference of a circle and lay it out flat like a line, the line will be slightly more than 3x the diameter of the same size circle. If you do the math out properly, that slightly more than 3x diameter is exactly π.
The next time you buy a pizza, measure de diameter and divide it by 2, cut it in narrow slices, 20 for example. Grab them and put them side by side, alternating one with the crust towards you and the next with the crust opposite, measure the crust length on one side and multiply it by two (to get the length of both sides). It will be 3.14 times the radius. If your pizza was 15 cm radius, you’ll have ~47 diameter/length.
By arranging the slices that way you’re measuring the circumference in an easier, linear, way.
The more slices you have, the more perfect the circle of the pizza is made, the more accurate your result will be. They needed a way to measure the surface area and the volume of round objects and for that you need to know the ratio of the radius to the circumference which is called pi. It will always be 3.14 radiuses, no matter the size of your pizza. Like on a square, all the sides are the same length to each other, no matter how big or small the square is.
They saw that the line that goes across a circle’s center can go about 3.14 times around the whole circle
You can do this yourself! Get some string and a ruler and go around measuring the outside (circumference) of different circles you find, and then measure across the middle (diameter). If you divide the circumference by the diameter, you’ll find that they’re all about 3 and a bit.
Measuring more precisely will get you closer to π.
Ancient Greek philosophers were incredibly clever mathematicians. They asked really smart questions like “what is the ratio of a circle’s circumference to its diameter?”
Well they didn’t have fancy math tools like calculus or computers but they didn’t have a compass, a straight edge, and a lot of time to think. Since they couldn’t directly measure a curve like we can now, they took to estimating it. They figured out very quickly that all circles seem to have the same ratio. They simply called this ratio “pi”.
The ancient Greeks came up with a plan. If all you have is a compass and straight edge all you can draw are circles and lines. So you start building easy shapes like triangles and squares and hexagons oh my! You can “squeeze” a circle between 2 polygons, one larger and one smaller, so that their easy to calculate perimeters and diameters give you upper and lower bounds on the circle’s circumference. If you set the diameter equal to 1 then the circumference must itself be pi. Using this method they got 3.16 which ain’t too bad for 2500 years ago.
The problem is that we don’t even know if pi could be written as a fraction. It wasn’t until the development of what we might call “modern” mathematics did we prove that pi is irrational (can not be written as a fraction) and then awhile later we proved that it must be transcendental (can not be the solution to a special family of functions called polynomials).
So we can’t write it as a fraction and it can’t be found from standard equations you might learn in school so what can we do? Well we can do what the ancient Greeks did but better because we have supercomputers now. We have come up with incredibly fancy ways of calculating estimations of pi to trillions of decimals and there’s still more we don’t know! For example, we don’t know if pi is a normal number. This would mean that if you counted how many times each numeral appears (how many 1s you see, how many 2s, 3s, 4s, etc.) you should get them all appearing roughly the same number of times.
Ignoring the actual question, lemme toss in a plug for one of my favorite books, “A History of Pi” by Petr (not a typo) Beckmann.
Includes bonus extended rant on the evils of the Roman Empire.
Take circle where you know the circumference.
The start plugging random numbers into the place of pi if the answer you get is too big than make that number smaller if it’s too small than make the number bigger. So…
3 too small
4 too big
3.5 too big
3.2 too big
3.1 too small
3.15 too big
3.14 too small
And continue doing that forever until you get as many digits as you want.
I think a visual representation is best for ELI5: https://www.youtube.com/shorts/xeZU4oWO6p0?feature=share
(turn the audio off or down)
What I think is funny, when the ancient geometry/math guys did this, how pissed off were they that not only did it not fit a nice round number, but it didn’t fit any number?
Or was that more exciting for them?
The cool thing about math is seemingly unrelated topics can often be used to prove universal truths.
So while others have given great answers about the more simple mathematical proofs, here’s an experiment to calculate pi with randomness and darts.
https://youtu.be/M34TO71SKGk
Archimedes of Syracuse (that’s a Greek fella from over 2250 years ago) was the first jack to write it down that we know of.
He drew a circle and broke it down into straight lines, like drawing a pentagon, hexagon, etc around a circle.
Turns out, the radius averages out to coincide with the circumference. And the more angles it takes, (like going from a hexagon to an octagon) the more accurate the ratio for ‘Pi’.
The ancient Greeks were pretty good at writing shit down, though. So, it’s possible that this “ratio” had been discovered previously by mathematicians from other areas and or eras.
They are called Mathematicians and scholars. Not that many of them around anymore.
According to my calculus professor years ago, if we were to find the final number of pi, we would be able to put a round roof on a square building with zero overhang from either object.
Pi is a number, you can’t “come up with it” any more than you could with for example two, or three.
After reading the comments…I didn’t even know pi was for a circle
A bunch of great answers here.
Now try and measure a coastline. ; )
It was discovered by a hard working pizza guy working late at night in Italy
https://www.youtube.com/shorts/pgGrQjYcm5A
it is a universal given. when/if we meet intelligent aliens, we’ll start with pi and go from there.
https://youtu.be/gMlf1ELvRzc?si=dKIwUJa-uXtJG6Pg
Good video here from Veritasium on the history of how Pi was calculated going back to ancient times.
I can’t remember all the details, but one of thr most impactful college courses i took started with some lectures on how ancient civilizations estimated Pi. All around the same time, you got estimates of 3.16, 22/7, etc. By various methods, they were all somewhat close, one being as close as 3.1416 in ancient India if I remember correctly.
Most notably, an ancient East Asian culture estimated it by comparing the longest chord of a regular polygon with the measure of its perimeter. I think they were able to get up to a 17 sided polygon before being unable to calculate further, and correctly had the first 7 digits of pi! It was so crazy because each culture was close, but this one was so incredibly accurate
They calculated the sum of the sides of a square with a diagonal length of 2 (each corner is 1 from the center point in all of these), then a pentagon, then a hexagon, then a septagon, an octogon, and on and on until they could see a pattern. As the number of sides increased, the closer the answers got. They called the answer that they would finally reach infinite, a circle, sides as pi. Later, some very smart people with a ton of time on their hands and the invention of calculus they were able to figure out the actual value.
This guy did a really good video on this:
https://www.youtube.com/watch?v=gMlf1ELvRzc
Ug was a caveman. He was bored.
Ug had a vine which he wrapped around a tree for fun. ( He was very bored)
He noticed that the length of vine that went around the tree once was the same as about three times the width of the tree.
The same thing happened when he tried it on a different tree.
Huh, that’s weird, thought Ug.
Doesn’t matter how big the tree was, the length around it was always about three times the width.
That’s π.
How is this taught in every single school in the world? You have enough intelligence to access the internet and type the question on reddit, but never learned how circles work? What?
1 Kings 7:23 23
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it. 24 Below the rim, gourds encircled it—ten to a cubit. The gourds were cast in two rows in one piece with the Sea.
2 Chronicles 4:2
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits[b] high. It took a line of thirty cubits[c] to measure around it.
https://youtu.be/gMlf1ELvRzc?si=bCArBOFb1EAQSKLQ
This video is godlike at explaining Pi but also how Newton massively changed the calculation approach. It’s long but fascinating
This was shown to us in gradeschool.
It is a decent history of math in a nutshell over 1 hour.
https://youtu.be/wMOiErUqrpM