Math question re Pi

[bekkilyn]

Forget the 2 factor. Pi would be c/r =6.28ish.

And how would you mathematically prove that Pi is really the ratio of the circumference to the radius instead of the ratio of the circumference to the diameter? Especially since the radius cannot be equal to the diameter.

 

[durangodawood]

Pi x r/squared is the formula for the area of a circle.

Pi x D is the formula for the circumference of a circle.

Two entirely different measurements.

???
D is just 2r. They are completely interchangeable for the purpose of deriving Pi.

 

[durangodawood]

And how would you mathematically prove that Pi is really the ratio of the circumference to the radius instead of the ratio of the circumference to the diameter?....

Because D is precisely equal to 2r. The conversion could not be more simple.

 

[durangodawood]

Convention has us express things in the simplest terms possible. We could say that Pi = C/2r, but since 2r = d, it is simpler to say that Pi = C/d. Similarly, we could say that A = Pi * (d/2)^2, but it is simpler to say that A = Pi * r^2.

In short, we express things as simply as possible. In some contexts that means we use r, and in some contexts we use d; whichever is simpler.

In terms of r Pi =6.28... is exactly as simple as Pi =3.14... in terms of D. The complexity either way is identical.

 

[bekkilyn]

Because D is precisely equal to 2r. The conversion could not be more simple.

You said to forget the 2 and thus change Pi to 6.28ish. How would you mathematically prove that 3.14ish is equivalent to 6.28ish and that r is equal to 2r since the 2 factor doesn't matter?

If we forget the 2, then we no longer have c/(2r) but instead c/r which is not at all the same formula.

 

[bekkilyn]

In terms of r Pi =6.28... is exactly as simple as Pi =3.14... in terms of D. The complexity either way is identical.

If you divide the denominator by 2, then you must divide the numerator by two, so to get rid of the 2 in the denominator, you would need to write the ratio as (c/2)/(2r/2) which simplifies to (c/2)/r which then simplifies right back to c/(2r).

 

[Knee V]

In terms of r Pi =6.28... is exactly as simple as Pi =3.14... in terms of D. The complexity either way is identical.

I see what you're saying now.

Maybe the diameter is more of a "real thing" than the radius?

 

[OldWiseGuy]

Not entirely different, since D = 2r.

I meant the measurements have different purposes. For instance the circumference of a barrel isn't necessary to calculate it's volume.

 

[Sabertooth]

For instance the circumference of a barrel isn't necessary to calculate it's volume.

It would be, if you knew where A = pi x r^2 came from.

Area of a Circle

 

[OldWiseGuy]

It would be, if you knew where A = pi x r^2 came from.

Area of a Circle

Why make it more complicated than it needs to be?

Some people use a hammer spinning awkwardly through the air to demonstrate that the center of balance of such an object forms a smooth ark.

I use it to drive nails.

 

[Sabertooth]

Why make it more complicated than it needs to be?

That is where A = pi x r^2 came from. It riffs on circumference. (Knowing that, we just commit it to memory.)

Your barrel calculation still uses circumference in the process. It is just contained within the rote formula.

 

[Occams Barber]

Why is the convention to express Pi in terms of the diameter rather than the radius? Arbitrary? Some good reason?

I'm used to seeing circumference expressed as both *D and 2*R. If anything, 2*R is the more conventional.

Then again I do live in an upside-down country.

OB
(Edit: *Oooops - my pasted Pi's have fell off)

OB

 

[Occams Barber]

I meant the measurements have different purposes. For instance the circumference of a barrel isn't necessary to calculate it's volume.

How can you calculate the volume of a barrel without using the circumference?

OB

 

[essentialsaltes]

It is entirely arbitrary.

The use of the Greek letter pi for this ratio is really only a few centuries old. But once that particular standard was chosen and populartized, the arbitrary choice became fixed.

The concept of a fixed ratio for circles is thousands of years older and developed independently in several cultures. Some defined it differently.

 

[OldWiseGuy]

How can you calculate the volume of a barrel without using the circumference?

OB

Volume = height x Pi x r squared.

No circumference (Pi x D) needed.

 

[Sabertooth]

Volume = height x Pi x r squared.

No circumference (Pi x D) needed.

Pi comes from c/d.

Without pi, volume = height x (circumference/diameter) x (radius, squared)

 

[OldWiseGuy]

Pi comes from c/d.

Without pi, volume = height x (circumference/diameter) x (radius, squared)

But we don't need to reinvent Pi, it's a 'given'.