# Why is pi = circumference / diameter ?

• Juwane
In summary, pi is a 2500 year old convention that was likely based on the idea of expressing the perimeter of a figure as a multiple of a unit length. It is the ratio of the circumference to the diameter, and this convention has remained unchanged despite the possibility of defining it differently. Pi is also the first letter in the word "perimeter," which may have influenced its adoption.
Juwane
Why do we take pi as the ratio of the circumference to the diameter, and not diameter to the circumference? Is it because circumference is always bigger than the diameter, so that it will be easy to work with the ratio? Or is it something fixed by those who discovered it and we can't change it?

You also should realize that this is the ratio of the circle's area to the square of its radius. That is, a unit circle has an area of pi.

It's just a definition...

What if we turn it upside down; that is, diameter over circumference. This is still a ratio, isn't it?

Juwane said:
What if we turn it upside down; that is, diameter over circumference. This is still a ratio, isn't it?

That would be perfectly acceptable as well. pi is just a number someone defined (probably someone from ancient Greece), but you can define lots of other numbers related to it if you want. At some point we defined pi and since then we haven't found a major reason why we would rather have diamter/circumference or anything else related to it. In some areas of math we more often work with $\sqrt{\pi}$, $2\pi$ or $1/\pi$ than $\pi$, but we don't re-define pi or introduce a new symbol since it wouldn't be worth the trouble.

Why do we take pi as the ratio of the circumference to the diameter, and not diameter to the circumference? Is it because circumference is always bigger than the diameter, so that it will be easy to work with the ratio? Or is it something fixed by those who discovered it and we can't change it?

It's a 2500 year old convention.

You may as well ask why pi and not rho

Juwane said:
Why do we take pi as the ratio of the circumference to the diameter, and not diameter to the circumference? Is it because circumference is always bigger than the diameter, so that it will be easy to work with the ratio? Or is it something fixed by those who discovered it and we can't change it?

In ancient times geometers probably thought of the perimeter of a figure a derived or constructed from a unit length and so expressed the perimeter as a multiple of one. A circle is well approximated as a perimeter of a many sided regular polygon.

From this point of view the unit length would be the radius of the circle so I would guess that the original formula was circumference/2xradius.

Borek said:
You may as well ask why pi and not rho

Because it's the first letter in the word "perimeter".

That still puts us in the world of conventions. What if pi was defined earlier by Egyptians? Or later by Muslims? It would be as obvious that we use some different symbol as it is that we use pi now.

I don't know, why do we call a chair a "chair" and not a "kanildor?"

Convention

## 1. What is pi?

Pi is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14159, but it is an irrational number, meaning it has an infinite number of digits after the decimal point.

## 2. How is pi calculated?

Pi is calculated by dividing the circumference of a circle by its diameter. It can also be approximated using various mathematical formulas and methods, such as the infinite series or the Monte Carlo method.

## 3. Why is pi important?

Pi is important in mathematics and science because it is a fundamental constant that appears in many equations and formulas relating to circles and spheres. It is also used in calculations involving angles, trigonometry, and calculus.

## 4. Can pi be calculated accurately?

No, pi cannot be calculated accurately because it is an irrational number. This means that it has an infinite number of non-repeating digits after the decimal point, making it impossible to represent it as a finite decimal or fraction.

## 5. What is the history of pi?

The earliest known calculations of pi can be traced back to ancient civilizations such as the Babylonians and Egyptians. The Greek mathematician Archimedes is credited with calculating its value to be between 3.1408 and 3.1429. The symbol "π" was first used to represent pi by William Jones in 1706, and it has been studied and calculated by mathematicians and scientists ever since.

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