Can a Circle be Described as a Polygon?

[Universe_Man]

Recently, due to my newfound obsession with circles and Pi, I came across an idea that may or may not hold up, and I need some feedback on it.

It occurred to me that a polygon with a sufficient number of angles could closely resemble a circle. For example, let's say you have a regular polygon with 1 billion sides. each interior angle is roughly 179.999 degrees, which makes the angles almost unnoticeable at all unless you have an implement of measuring. If you actually had a drawing of a billion sided regular polygon, it would look like a circle unless you measured it. I have also come to the conclusion that it would make sense that the any single interior angle can never be exactly 180 degrees, as it would be a straight line. You could get extremely close to 180 degrees, but never reach it.

To tie this in more with the circles, I assumed that perhaps a circle could be loosely described as a polygon with an infinite number of sides. Perhaps I am wrong in this assumption, but it kinda makes sense to me.

I apologize if this is common math knowledge or if I am wrong, I just like to throw ideas that I think of during the day around.

 

[Curious3141]

Recently, due to my newfound obsession with circles and Pi, I came across an idea that may or may not hold up, and I need some feedback on it.

It occurred to me that a polygon with a sufficient number of angles could closely resemble a circle. For example, let's say you have a regular polygon with 1 billion sides. each interior angle is roughly 179.999 degrees, which makes the angles almost unnoticeable at all unless you have an implement of measuring. If you actually had a drawing of a billion sided regular polygon, it would look like a circle unless you measured it. I have also come to the conclusion that it would make sense that the any single interior angle can never be exactly 180 degrees, as it would be a straight line. You could get extremely close to 180 degrees, but never reach it.

To tie this in more with the circles, I assumed that perhaps a circle could be loosely described as a polygon with an infinite number of sides. Perhaps I am wrong in this assumption, but it kinda makes sense to me.

I apologize if this is common math knowledge or if I am wrong, I just like to throw ideas that I think of during the day around.

It makes perfect sense. The limit of an n-gon as n tends to infinity is a circle.

You will find this interesting : http://staff.washington.edu/skykilo/Pi/Pi.html

 

[bindwood]

very good explanation!
tnx for your sharing

 

[DaveC426913]

To tie this in more with the circles, I assumed that perhaps a circle could be loosely described as a polygon with an infinite number of sides. Perhaps I am wrong in this assumption, but it kinda makes sense to me.

More than "loosely" - I have heard this as a decription of a circle many times.

 

[mathman]

Archimedes (if I remember correctly) estimated pi by approximating a circle with a polygon with many sides.

 

[masudr]

You do remember correctly.

 

[arildno]

In fact, Archimedes used 96-gons.

UniverseMan:
Here's how Archimedes in essence proved that the area of a circle* is $\pi{r}^{2}$
where $\pi$ is the ratio between the diameter D and circumference C of the circle (i.e, we have $\pi=\frac{C}{D}$ and r the radius of the circle.

Now, consider a regular N-gon with total circumference L that is inscribed in a circle; the vertices of the N-gon lying on the circle.

Draw lines from each vertex to the center of the circle; and consider the area of a single of those triangles thus created. Setting its height to h, we clearly have that a triangle's area is $\frac{hL}{2N}$, and therefore, the N-gon's total area must be $N*\frac{hL}{2N}=\frac{hL}{2}$ since the N-gon consists of N such triangles.

Now, as N gets big, h will tend to the radius r, and L will tend to the circle's circumference C; thus, the N-gon's area should approach the value:
$\frac{rC}{2}=\frac{rD\pi}{2}=\pi{r}^{2}$
which is what we reasonably can be called the circle's* area..




*More precisely, the area of the solid disk with the circle as its boundary curve.

 

[Universe_Man]

Hey, I thought I would try to add something more to my thread here. This is probably well known, or already thought of most certainly, but I would just like to confirm my own personal exploration.

Let's say you were to find Pi by calculating the area of two polygons inscribed and circumscribed around a circle like the ancient Greeks did. the more sides your two polygons have, the more accurate your calculation of Pi would become.

Since a circle can be described as a polygon with an infinite number of sides you can calculate as many sides as you want to, always coming closer and closer to a circle. That's why Pi has an infinite number of decimal places.

 

[StatusX]

That's not quite right. By an infinite number of decimal places, I assume you mean a non-repeating decimal expansion, and the numbers that have such expansions are precisely the irrational numbers, ie, those numbers which are not equal to p/q for any integers p and q. It is true that pi is irrational, but this is not because it can be approximated by a series (of areas of polygons in this case). For example, the series 1, 1+1/2, 1+1/2+1/4, ... gets closer and closer to 2, but 2 is rational. It may be possible to extend the approximation of pi by polygons with arbitrarily large numbers of sides into a proof of the irrationality of pi, but the fact that pi can be approximated this way is not enough by itself. If you want a proof of the irrationality of pi, try http://www.lrz-muenchen.de/~hr/numb/pi-irr.html .

 

[chroot]

Since a circle can be described as a polygon with an infinite number of sides you can calculate as many sides as you want to, always coming closer and closer to a circle. That's why Pi has an infinite number of decimal places.

This is false. There are many limit processes that do not result in irrational numbers.

- Warren

 

[DeadWolfe]

The fact that a circle is a polygon with infinite sides features heavily in the plot of flatland by the way...

 

[mathwonk]

it would be true if each of your approximating polygons had an area given by a finite decimal.

 

[Robokapp]

Let me dig out my old post about this...it's coming from another forum so it's not for math people but for...general audience so it's bumbed down a little but...i hope it makes sense.

"Pi being rational especially. my next paragraph will be a disorganised mess of math never before produced in full sentences at 6 am but here I go:

assuming a circle is a polylateral figure with infinity sides, its area can be described as infinity times zero times R/4. I get that from the formula "number of sides times apothem times side divided by two" also known shortly as "Semiperimeter times apothem". Formula works for any figure with straight sides. I'll consider a cirle a polylateral figure with sides of zero length...and infinity in number.

So...apothem is haf the radius...and semiperimeter would be infinity divded by two...i'll move the two with the other one to form a 4...etc. What is my point?

Well, infinity times zero is a Calculus mess...because it's infinitely decimalled. yes, infinity times zero does have an answer...and it's neither infinite, nor zero, nor is the same every time. infinity is not a number but a concept.

A circle is this way described as having its surface much like that of a fractal. That is a figure that the closer you look the more details and curves you notice. It's a figure bound by a line that can't be analised fully because as you go to a smaller decimal place, more change exists, and more decimal places are needed.

That's my approach on the concept of Pi being irrational. Area of a circle equals Pi times R^2 and the R is no surprise at all, therefore the PI must be the one carrying the infinite decimals.
"

~Robokapp

 

[darius]

Archimedes (if I remember correctly) estimated pi by approximating a circle with a polygon with many sides.

ditto! I have been reading Archimedes and it is amazing how advanced the Greeks got with math!

 

[Alkatran]

Let me dig out my old post about this...it's coming from another forum so it's not for math people but for...general audience so it's bumbed down a little but...i hope it makes sense.

"Pi being rational especially. my next paragraph will be a disorganised mess of math never before produced in full sentences at 6 am but here I go:

assuming a circle is a polylateral figure with infinity sides, its area can be described as infinity times zero times R/4. I get that from the formula "number of sides times apothem times side divided by two" also known shortly as "Semiperimeter times apothem". Formula works for any figure with straight sides. I'll consider a cirle a polylateral figure with sides of zero length...and infinity in number.

So...apothem is haf the radius...and semiperimeter would be infinity divded by two...i'll move the two with the other one to form a 4...etc. What is my point?

Well, infinity times zero is a Calculus mess...because it's infinitely decimalled. yes, infinity times zero does have an answer...and it's neither infinite, nor zero, nor is the same every time. infinity is not a number but a concept.

A circle is this way described as having its surface much like that of a fractal. That is a figure that the closer you look the more details and curves you notice. It's a figure bound by a line that can't be analised fully because as you go to a smaller decimal place, more change exists, and more decimal places are needed.

That's my approach on the concept of Pi being irrational. Area of a circle equals Pi times R^2 and the R is no surprise at all, therefore the PI must be the one carrying the infinite decimals.
"

~Robokapp

...
There are so many thing wrong with that post. :uhh: I feel sorry for the people you confused with it. Main mistake: not realizing you need to use a limit.

 

[Robokapp]

:( Well I tried. That's the best I could come up with at...that hour. Yeah...sry for intoxicating you with what I only think it's math. I do that quite often.

It's a bad habit that I can't get rid of. if I think it's real, and if it makes sense to me...well how can I not consider it real?

 

[Alkatran]

:( Well I tried. That's the best I could come up with at...that hour. Yeah...sry for intoxicating you with what I only think it's math. I do that quite often.

It's a bad habit that I can't get rid of. if I think it's real, and if it makes sense to me...well how can I not consider it real?

It's fine to think like that, just don't do proofs like that! The general population already has a bad enough idea of what math is.

 

[BoTemp]

A bit clearer

assuming a circle is a polylateral figure with infinity sides, its area can be described as infinity times zero times R/4. I get that from the formula "number of sides times apothem times side divided by two" also known shortly as "Semiperimeter times apothem". Formula works for any figure with straight sides. I'll consider a cirle a polylateral figure with sides of zero length...and infinity in number.

~Robokapp

More mathematically, take a regular polygon with n sides, and distance from the center to halfway between any side as r (aka radius). The area of this polygon can be found by breaking the polygon into right triangles, and comes out to be n*r^2*sin(pi/n).

$$\lim_{n\rightarrow\infty} nr^2sin(\pi/n) = \pi r^2$$

So yes, it's infinity times zero, but in rigorous mathematical terms it's taking the limit of an expression. Infinity times 0 is not defined. Ever. What may be defined is the limit of an expression which contains the product of two expressions whose limits are infinity and 0.

 

[Robokapp]

Correct...it's as dividing area under a curce in rectangles...closer to infinity means a width closer to zero but you don't end up that far ever.

However...it wasn't a math forum, I couldn't trow in the word "limit". I hope you understand. i first heard of limits (the math limits of course) in Calculus AB so...I doubt I'd be understood.

But I see what you mean. The actaul article was about wheather or not Pi was rational by the way...but I ommited my last paragraph.

 

[Universe_Man]

Hey everyone, thanks for helping out.