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 April 9th, 2018, 04:23 PM #1 Global Moderator   Joined: May 2007 Posts: 6,768 Thanks: 699 circle area The ratio of the circumference of a circle to its diameter (2r) is given by $\pi$ (definition). The area of a circle is $\pi r^2$. Can the area be derived without calculus?
April 9th, 2018, 04:29 PM   #2
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 Originally Posted by mathman The ratio of the circumference of a circle to its diameter (2r) is given by $\pi$ (definition). The area of a circle is $\pi r^2$. Can the area be derived without calculus?
Archimedes did so in 260BC.

https://en.wikipedia.org/wiki/Area_o...es&#39;s_proof

http://www.ams.org/publicoutreach/fe...umn/fc-2012-02

 April 9th, 2018, 04:34 PM #3 Senior Member     Joined: Sep 2015 From: USA Posts: 2,463 Thanks: 1340 Are limits calculus? The limit of the area of a regular n-gon with "radius", (i.e. center to vertex length), $r$ has area $A_n = n \sin^2\left(\dfrac \pi n\right) \cot\left(\dfrac \pi n\right)$ and $\lim \limits_{n\to \infty}~A_n = \pi r^2$
April 9th, 2018, 07:55 PM   #4
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 Originally Posted by Maschke Archimedes did so in 260BC. https://en.wikipedia.org/wiki/Area_o...es's_proof AMS :: Feature Column :: Measurement of a Circle
It seems to me that Archimedes proof (as presented in your link) explicitly uses calculus. The fact that he didn't realize the thing he was computing was a limit is certain. However, that is exactly what is being described in the 2nd link. In fact, I would say he proved it using the squeeze theorem.

Though I admit I don't know any proof which doesn't require some kind of limit so I would be really interested if someone presented something.

April 9th, 2018, 09:13 PM   #5
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 Originally Posted by SDK It seems to me that Archimedes proof (as presented in your link) explicitly uses calculus. The fact that he didn't realize the thing he was computing was a limit is certain. However, that is exactly what is being described in the 2nd link. In fact, I would say he proved it using the squeeze theorem.
Well sure, and Eudoxus invented integration. But calculus typically means the organized and formalized body of theory and technique that started with Newton and Leibniz. So now it comes down to what we mean by the word calculus. The underlying ideas are very old.

April 9th, 2018, 09:57 PM   #6
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 Originally Posted by Maschke Well sure, and Eudoxus invented integration. But calculus typically means the organized and formalized body of theory and technique that started with Newton and Leibniz. So now it comes down to what we mean by the word calculus. The underlying ideas are very old.
I can agree that the important thing becomes "what does without calculus" mean. I would consider a limit to be using calculus, even if the original author didn't refer to these as limits, or justify them rigorously, etc. In the end, this is ultimately a completely subjective question. Nevertheless, I would be excited to see a proof for the area of a circle which did not require overtly taking any limits.

Quote:
 Originally Posted by v8Archie You are obviously going to need the concept of a limit because π is irrational. You must therefore work in the real numbers (as opposed to the rationals) and the reals are limits of rational sequences (there are alternative but equivalent definitions). Any finite sequence of computations isn't going to give you $\pi$.
I disagree. We can talk about lots of irrational numbers without referring to Cauchy sequences or limits. For example, we can prove that $\sqrt{2}$ is irrational but this doesn't require calculus.

Even the definition of $\pi$ as the ratio of the circumference to diameter doesn't require calculus since you can prove this ratio is constant for any circle using just geometry.

I think the part of this question that is intimately tied to calculus is not the irrationality of $\pi$, but rather the fact that the notion of area itself is intimately tied to calculus. For example, you can choose an ellipse with rational axes and rational perimeter, but I still don't know how to compute its area without calculus.

April 10th, 2018, 12:56 AM   #7
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 Originally Posted by mathman The ratio of the circumference of a circle to its diameter (2r) is given by $\pi$ (definition). The area of a circle is $\pi r^2$. Can the area be derived without calculus?
Sure, how would you define area without calculus? And length?

Last edited by Micrm@ss; April 10th, 2018 at 01:00 AM.

April 10th, 2018, 01:09 AM   #8
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 Though I admit I don't know any proof which doesn't require some kind of limit so I would be really interested if someone presented something.
Although he doesn't quite do that one Mark Levi tackles something more ambitious in his delightful book

Quote:
 Here is a nonstandard solution of a standard calculus problem: Find the dimensions of the of a circle and square of given combined perimeter with the smallest combined area
The Mathematical Mechanic

Using Physical Reasoning to solve (Mathematical) Problems

 April 10th, 2018, 04:32 AM #9 Global Moderator   Joined: Dec 2006 Posts: 20,747 Thanks: 2133 I think romsek meant that a regular n-gon of "radius" $r$ has area $A_n = nr^2\!\sin^2\left(\dfrac \pi n\right) \cot\left(\dfrac \pi n\right)$.
April 10th, 2018, 04:36 AM   #10
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Quote:
 Originally Posted by SDK I disagree.
Me too, which is why I deleted my post before you posted your reply.

I think the transcendental nature of $\pi$ might fit the bill though.

Quote:
 Originally Posted by SDK I think the part of this question that is intimately tied to calculus is not the irrationality of $\pi$, but rather the fact that the notion of area itself is intimately tied to calculus. For example, you can choose an ellipse with rational axes and rational perimeter, but I still don't know how to compute its area without calculus.
Area does not require calculus: see a square, for example. We can define area perfectly well without calculus. It is only certain types of area that require calculus: specifically those with curved boundaries. And the only reason we require it then is that our (only) method involves taking the limit of polygons.

Last edited by v8archie; April 10th, 2018 at 04:44 AM.

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