Is Pi Interpreted Differently in Circle Area Calculation and Angle Measure?

[Adel Makram]

The area of a circle is πr². Here π is the constant that represents the ratio of the circumference to the diameter. But in deriving πr², 2π is used as an angle, which is the upper bound of the integral ∫½ r² dθ. So how the same π is used in 2 different meaning?

 

[axmls]

Area of a circle:

A = πr². π is the number that makes that equation true.

That's the definition my calculus 1 professor gave the class.

 

[Mentallic]

The area of a circle is πr². Here π is the constant that represents the ratio of the circumference to the diameter. But in deriving πr², 2π is used as an angle, which is the upper bound of the integral ∫½ r² dθ. So how the same π is used in 2 different meaning?

It's not two different meanings at all. If you instead used \tau=2\pi (pronounced tau) which is in most ways a more appropriate circle constant, then the upper bound to the integral would be \tau, the circumference would be C=\tau r and the area of the circle would be A=\frac{1}{2}\tau r^2.

The definition of $\pi$ being the ratio of the circumference to the diameter was useful back in the day when diameters were used most often since they were easier to measure than the radius is the only reason $\pi$ is still being used today. It's hard to change from the old.

Anyway, rant over, back to your question. The circle has a circumference $C=2\pi r$ which means that there are $2\pi$ radians in a full revolution of a circle. This is the reason your upper bound is $2\pi$. Just because the constant coefficient of the area of a circle is $\pi$ has no bearing on your integral. Similarly if you calculate the volume of a sphere, you won't be using $\frac{4}{3}\pi$ as your upper bound.

 

[Adel Makram]

The circle has a circumference C=2πrC=2\pi r which means that there are 2π2\pi radians in a full revolution of a circle. This is the reason your upper bound is 2π2\pi

So π means it corresponds to 180 degree but not equal to 180 degree in absolute sense.

 

[Mentallic]

So π means it corresponds to 180 degree but not equal to 180 degree in absolute sense.

$\pi$ is just a number like 1, -23 or $\sqrt{2}$. If we want to express length, then we have to include units, such as 1cm, -23ft or $\pi$ light years. But when doing geometry and every number corresponding to a side length represents a certain unit length (and it usually doesn't matter whether it's cm, ft, or ly) then we either add the unit u to represent any unit length, or more conveniently, we ignore it altogether and just assume it's a unit length.

Similarly for radians, we're expected to include units. The symbol for radians are either ^c or rad, so to represent 180^o we write either $\pi ^c$ or $\pi \text{ rad}$. But again just like before, when we're always dealing with angles, since radians are most commonly used then we can just exclude the radian symbols and it would be implied that they're radians. We don't want to mix this up with degrees though, so that's one reason why we always include the degree symbol (at least one of them has to always be included). No symbol hence implies radians.

Radians are an angle measure, and degrees are too, so $\pi^c \equiv 180^o$. They are the same thing just as 2.54cm = 1" (2.54 centimetres = 1 inch).

 

[stedwards]

The area of a circle is πr². Here π is the constant that represents the ratio of the circumference to the diameter. But in deriving πr², 2π is used as an angle, which is the upper bound of the integral ∫½ r² dθ. So how the same π is used in 2 different meaning?

Adel. You've asked a very subtle question that involves dimensionality as well as units, that is not in the usual habitate of typical mathematical concern. Good for you. You're thinking.It's a good idea not to confuse units and dimensionality of a quantity. For instance, radians per second has dimensions of [R/T] but units of [1/T]. (Most people don't make the distinction.)

First, let us define $\pi$ as the ratio of the circumference of any circle divided by twice its own radius. It has dimension of length over length, or $[L/L]$ and units of length of length, $[L/L]$.

Now, which integral for the area you asking about? There is more than one.

 

[Adel Makram]

Now, which integral for the area you asking about? There is more than one.

If I divide the circle into n triangles each has a small but finite base = r Δθ and sides r, then the area of this triangle is ½ r² Δθ. The area of the circle i=∑½ r² Δθ. When Δθ->dθ, ∑->∫. So the area of the circle =∫½ r² dθ with the upper bound is 2π and the lower one is zero.. The area then= πr².

 

[EM_Guy]

So π means it corresponds to 180 degree but not equal to 180 degree in absolute sense.

Forget degrees. That there are 360 degrees in a circle is an arbitrary designation. We could have just as easily said, "Let a circle be divided into 6024 equiangular arcs, and let each of these angles be called a "degree."

 

[EM_Guy]

If I divide the circle into n triangles each has a small but finite base = r Δθ and sides r, then the area of this triangle is ½ r² Δθ. The area of the circle i=∑½ r² Δθ. When Δθ->dθ, ∑->∫. So the area of the circle =∫½ r² dθ with the upper bound is 2π and the lower one is zero.. The area then= πr².

This is very cool. But how do we know that $C = 2\pi r$?

 

[EM_Guy]

Or is it that $\pi$ is the number that makes $C = 2\pi r$ true?

 

[micromass]

Area of a circle:

A = πr². π is the number that makes that equation true.

That's the definition my calculus 1 professor gave the class.

Not really a good definition since it needs to be shown that the constant is independent of $r$. Who knows, a priori it might be the case that the area of the unit circle is $\pi$, but the area of a circle with radius $2$ is more like $A = 3\cdot 2^2$. In fact, that this is not the case is quite deep and depends crucially on the parallel axiom: it is false on a sphere or on a hyperbolic plane. In those contexts, $\pi$ should be seen as related to the area of an infinitesimal circle.

 

[HallsofIvy]

Actually it is more common to define \pi as the circumference of a circle divided by the diameter of the circle. After, of course, showing that "all circles are similar"- that is, that circumference divided by diameter is the same for all circles.

 

[Adel Makram]

This is very cool. But how do we know that $C = 2\pi r$?

I think proving c=2πr may be inconsistent. Because at one point of the proof, π=c/2r will be again used as an axiom which makes using an axiom to prove it inconsistent by Godel incompleteness theorem.

 

[pwsnafu]

I think proving c=2πr may be inconsistent. Because at one point of the proof, π=c/2r will be again used as an axiom which makes using an axiom to prove it inconsistent by Godel incompleteness theorem.

That's not what Godel incompleteness theorems (because there's two) say. More to the point, Euclidean geometry is proven to be consistent and complete, and Godel's theorems don't apply.

 

[Adel Makram]

That's not what Godel incompleteness theorems (because there's two) say. More to the point, Euclidean geometry is proven to be consistent and complete, and Godel's theorems don't apply.

I mean π is equal to the ratio of the circumference to the diameter by definition. Nothing to be proved here.

 

[Adel Makram]

That's not what Godel incompleteness theorems (because there's two) say. More to the point, Euclidean geometry is proven to be consistent and complete, and Godel's theorems don't apply.

Do you mean Euclidean geometry can prove any of its 5 axioms?

 

[pwsnafu]

Do you mean Euclidean geometry can prove any of its 5 axioms?

It's important to understand in axiomatic set theory, "axiom" and "proof" are technical terms. The general mathematician's interpretation that "axioms cannot be proved" is inadequate. For a given theory (such as EG or PA or ZF), a subset of all sentences (technically called "well formed formulas") in the theory are designated as "axioms". A "formal proof" of a sentence is a finite sequence which ends with the sentence in question, and only contains axioms, assumptions or sentences which can inferred from previous sentences. This means, suppose "A" is an axiom, then one element sequence

A

is infact a formal proof of A itself. In this narrow setting, yes, the axioms of Euclidean geometry have proofs. Another way to look at this is to define axioms as those sentences with trivial proofs.

Godel's incompleteness theorems are technical results of Peano arithmetic, not Euclidean geometry. It turns out that arithmetic of the natural numbers has a structure that Godel's proof exploits, but that structure is not present in Euclidean geometry. Hence, Godel's theorems don't apply.

Secondly, Godel's theorems are not talking about axioms. Goodstein's theorem is an example of a true-but-not-provable-within-PA theorem. You'll notice it's very different from the axioms of PA.

 

[Adel Makram]

another way to look at this is to define axioms as those sentences with trivial proofs.

So, is c=2πr a trivial proof that c=2πr?

 

[micromass]

So, is c=2πr a trivial proof that c=2πr?

It is not usually taken as an axiom of Euclidean geometry.

 

[HallsofIvy]

It is proved in Euclidean geometry that the circumference of any circle is proportional to its radius. \pi is then defined as that constant of proportionality.

 

[Dr. Courtney]

It is proved in Euclidean geometry that the circumference of any circle is proportional to its radius. \pi is then defined as that constant of proportionality.

https://en.wikipedia.org/wiki/Euler's_identity

The basic definition is right and perfect. But Euler is beautiful!

 

[WWGD]

I hope I am not throwing the discussion too far off, but, is there a good reason for why $d/dr(\pi r^2) =2\pi r$ , i.e., the circumference as a function of $r$ describes the rate of change of the area as a function of $r$?

 

[micromass]

I hope I am not throwing the discussion too far off, but, is there a good reason for why $d/dr(\pi r^2) =2\pi r$ , i.e., the circumference as a function of $r$ describes the rate of change of the area as a function of #r#?

Haha, yes. It holds more generally than that too. It is a consequence of Stokes' theorem which connects the boundary of a manifold with the interior (but it isn't a straightforward consequence). I'll see if I can find the resources again.

 

[micromass]

OK, here is the main paper I was looking for. It details the most general law for "integration under the integral sign": http://sgpwe.izt.uam.mx/files/users/uami/jdf/proyectos/Derivar_inetegral.pdf Using this formula, you can see that the differentiation of the area of a circle is the length of the circle, and the same for hyperspheres and many other figures.

Other approaches can be found here: http://arxiv.org/pdf/math/0702635.pdf and here: http://blogs.sfu.ca/people/zazkis/wp-content/uploads/2010/05/Derivative-of-Area.pdf This shows that an equivalent relation holds also for cubes and other polygons.

 

[WWGD]

OK, here is the main paper I was looking for. It details the most general law for "integration under the integral sign": http://sgpwe.izt.uam.mx/files/users/uami/jdf/proyectos/Derivar_inetegral.pdf Using this formula, you can see that the differentiation of the area of a circle is the length of the circle, and the same for hyperspheres and many other figures.

Other approaches can be found here: http://arxiv.org/pdf/math/0702635.pdf and here: http://blogs.sfu.ca/people/zazkis/wp-content/uploads/2010/05/Derivative-of-Area.pdf This shows that an equivalent relation holds also for cubes and other polygons.

Excellent, thanks.

 

[Adel Makram]

https://en.wikipedia.org/wiki/Euler's_identity

The basic definition is right and perfect. But Euler is beautiful!

The shown diagram in wikipedia, I understood the end point of (1+iπ/N)^N as N increases but I didn`t get how the arc is divided into N segments?

 

[Adel Makram]

It is proved in Euclidean geometry that the circumference of any circle is proportional to its radius. \pi is then defined as that constant of proportionality.

I thought that c=2πr is a definition not a theorem. Can you please post a link of a proof?

 

[verty]

I hope I am not throwing the discussion too far off, but, is there a good reason for why $d/dr(\pi r^2) =2\pi r$ , i.e., the circumference as a function of $r$ describes the rate of change of the area as a function of $r$?

Another way to see it is to use a limit argument. The circumference is everywhere perpendicular to the radius, and in the limit as $\Delta{r} \to 0$, the shell region formed by rotating $r + \Delta{r}$ can be divided into tiny rectangles (they are trapezia but in the limit they are rectangles). One can see the change in area is the length of the circumference (by which I mean, ${dA \over dr} = C$).

 

[HallsofIvy]

I thought that c=2πr is a definition not a theorem.

No, the circumference of a circle is defined as the length of the curve making up the circle.

Can you please post a link of a proof?

Here is one proof: http://www.cpm.org/pdfs/state_supplements/
Here is the translation from Euclid's Elements. http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html