What is Pi?

1. Jun 16, 2013

vee6

I know a Pi is 3.14159265359 or 22/7.

Or Pi is a ration between a circle's circumference and its diameter.

Why Pi = 180 deg and 2 pi = 360 deg?

2. Jun 16, 2013

Simon Bridge

$\pi$

22/7 = 3.1429 i.e. it is not pi.
similarly pi is not 180 degrees.

Your second one is correct - pi is "Pythagoras' number" - so it carries the first letter of his name.
It is defined as the ratio of the circumference to the diameter of a circle.

22/7 is an old lower-school approximation for pi from before calculators were common - it is not used any more.

For a circle of radius R, an arclength of R subtends an angle, at the center, of 1 radien. This is the definition of a radien. Since there are 2pi radiuses in a full circumference, which is also 360 degrees, then one-pi radien must correspond to 180 degrees.

Note: 360 degrees to a circle means that the circumference of your protractor has to be 360 units. This means that the radius has to be an irrational number ... which can make building a very accurate one annoying. The bonus is that the numbers for common angles come out nice.

But if you choose radiens for your angle-measurements, it means that the radius is 1 unit long - which is easy to build. As a bonus there are lots of other simplifications as well - at the expense of having to keep that pesky pi in your calculations when you have simple angles. That turns out to be fine for scientific work since it often cancels out with a pi someplace else.

Last edited: Jun 16, 2013
3. Jun 16, 2013

lurflurf

So in angle measure units

or some people take degrees to be the constant

degrees=pi/180

then it is true that

pi=180 degees

4. Jun 17, 2013

vee6

What is the radian and pi relationship?

Pi is a pi.

Is pi a relative?

5. Jun 17, 2013

Simon Bridge

I gave you the answer to that in post #2.

A pi is not a radian ... a radian is an angle.

pi is just a number - you can have pi amount of anything you like.

You can have pi radians just like you can have pi degrees and pi centimeters and pi anything.
Because of the way radians are defined, an angle of pi radians happens to be the same as an angle of 180 degrees.

The radian is defined to be the angle that gives an arc-length equal to the radius of the circle.
You work it out.

If you make a wheel that has a radius of 1m, and you roll it along the ground without slipping, and you stop when the wheel has turned 180 degrees ... how many meters along the ground did the wheel move?

6. Jun 17, 2013

jonax101

Last edited: Jun 17, 2013
7. Jun 17, 2013

Simon Bridge

Hi jonax101, welcome to PF.
There's also the sorority from Revenge of the Nerds.

8. Jun 18, 2013

vee6

Which one is the constant?

Your above statement seems the radians is the constants, not the degrees.

9. Jun 18, 2013

vee6

There is a method to find the arc length of circle by integration.

Why need a pi to find the circumference of a circle?

10. Jun 18, 2013

Simon Bridge

@vee6:
The questions you are asking are very basic - what level of education do you have?

Neither degrees nor radians are "constant". Radians are just more useful in some situations than in others.
You can define units to be anything you like ... I could, for example, make the "unit" for angle the circumference of the circle ... then all angles will be less than 1.

There is no "why" for needing pi to find the circumference of a circle - it just is: there is no other way it could be and still be a circle. Just like the the square on the hypotenuse is always the sum of the squares of the other two sides or 1+1=2.

Put another way: if you can find the circumference of a circle without pi, and prove it, then you will become very famous.

However:
You do not seem to be paying attention to the answers you are being given: you keep repeating questions that have already been answered. Why should anyone bother answering you when you don't learn?

11. Jun 19, 2013

micromass

Right, and if you follow that method, then $\pi$ will appear.

12. Jun 19, 2013

marcusl

Not quite, you are being too casual with your numbers and units. The first line is correct. The second should be
1 degree = pi/180 radians.
The third line is then identical to the first.

As Simon Bridge already noted, there is no meaning to "degrees is the constant".

13. Jun 20, 2013

vee6

How come?

Please write an example.

14. Jun 20, 2013

HallsofIvy

In other words you do not know what the word "constant" means. $\pi$ is a number, about 3.1415962... (it is irrational and so cannot be written as a decimal number with a finite number of digits). That has nothing to do with "radians" or "degrees".

One application of $\pi$ (admittedly the most common) is in circle measurement. If you use "radians" to measure angles, then a "straight angle" would be measured as $\pi$ radians. If you used degrees, it would be 180 degrees. It is only in that sense that "$\pi$ radians is the same as 180 degrees". Asking "Which is the constant? Degrees or radians?" is like, after being told that 1 meter is (approximately) the same as 39 inches, asking "which is the constant, meters or inches?"

The word "constant" can only be applied to numbers or functions of numbers, not to "units" such as radians and degrees (or meters and inches).

15. Jun 20, 2013

vee6

Please write me an example, the integration to find the circle circumference that will makes the pi appears.

16. Jun 20, 2013

micromass

The circle is given by $(\cos(t),\sin(t))$. Then

$$\int_0^{2\pi} \sqrt{ x^\prime(t)^2 + y^\prime(t)^2 }dt = \int_0^{2\pi}\sqrt{\sin^2(t) + \cos^2(t)}dt = \int_0^{2\pi} dt = 2\pi$$

17. Jun 20, 2013

Simon Bridge

Is it your contention that it is possible to use integration to find the circle circumference where pi does not appear?
In that case it is up to you to demonstrate it.
Can you produce an example of what you mean by an integration method that does not make pi appear?
If so then please do so - if not then, where did you get this idea from?

In a trig function like sin(t) that "t" is an angle ... so the "pi" is in the definition of the angle: just like has been explained to you repeatedly in the previous answers. You don't seem to believe the previous answers so there is no reason to think you will believe anything else we can tell you.

It will be far more useful for us to help you with these confusions if you will show us your understanding rather than us just telling you what is true.

18. Jun 20, 2013

vee6

Why use sin t and cos t instead y = f(x)?

The above pi already appears (tex]\int_0^{2\pi}[/tex]) before it appears (2pi).

Wrong if you say "Right, and if you follow that method, then $\pi$ will appear."

Last edited: Jun 21, 2013
19. Jun 20, 2013

micromass

I don't see the objection, but if you want another method:

Take $f(x) = \sqrt{1 - x^2}$, then $f^\prime(x) = \frac{-x}{\sqrt{1-x^2}}$. Thus

$$\int_{-1}^1 \sqrt{1+ (f^\prime(x))^2}dx = \int_{-1}^1 \frac{1}{\sqrt{1 - x^2}}dx = arcsin(1) - arcsin(-1) =\pi$$

20. Jun 21, 2013

Simon Bridge

Having fun?
Which f(x) did you have in mind?

@Micromass: I don't think OP should get away with these vague statements: every example you try will have further objections:- nobody learns until they start doing their own work.

21. Jun 22, 2013

vee6

Still don't get the answer.

22. Jun 22, 2013

micromass

Can you say a bit more about what you don't understand and why not?

23. Jun 22, 2013

Simon Bridge

Then there is a communication problem - part of the problem is that you don't appear to want to say very much. For example: in the previous replies you have been asked a bunch of questions: you have yet to reply to any of them. These questions are not rhetorical - they are there to guide you (and us) to getting an answer you understand.

Please try to describe where we lose you - what is it you don't understand.
Don't worry about using the proper words or being clear - we get that you don't understand. We do understand ... we are used to people getting confused... and we are used to working through the confusion. Don't worry about looking silly: we've all been where you are - relax and gve it your best shot.

24. Jun 23, 2013

SW VandeCarr

Maybe a simpler approach. You accept that the circumference of a circle is $\pi$ times the diameter. The radius is one half the diameter. Therefore the circumference is $2\pi$ times the radius. We call the arc length equal to the radius and the angle associated with it a "radian". Therefore there are $2\pi$ radians in the full arc length of the circle (equal to the circumference) and $\pi$ radians in a the arc length of a half circle.

Last edited by a moderator: Jun 24, 2013
25. Jun 23, 2013

Simon Bridge

I have a feeling the persistent confusion lies in the way so many people think there is something mystical about pi.