April 9th, 2018, 04:23 PM  #1 
Global Moderator Joined: May 2007 Posts: 6,613 Thanks: 617  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  
Senior Member Joined: Aug 2012 Posts: 2,047 Thanks: 585  Quote:
https://en.wikipedia.org/wiki/Area_o...es's_proof http://www.ams.org/publicoutreach/fe...umn/fc201202  
April 9th, 2018, 04:34 PM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,124 Thanks: 1102 
Are limits calculus? The limit of the area of a regular ngon 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  
Senior Member Joined: Sep 2016 From: USA Posts: 476 Thanks: 263 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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  
Senior Member Joined: Aug 2012 Posts: 2,047 Thanks: 585  Quote:
 
April 9th, 2018, 09:57 PM  #6  
Senior Member Joined: Sep 2016 From: USA Posts: 476 Thanks: 263 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
Quote:
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 
Senior Member Joined: Oct 2009 Posts: 557 Thanks: 179  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  
Senior Member Joined: Jun 2015 From: England Posts: 887 Thanks: 266  Quote:
Quote:
Using Physical Reasoning to solve (Mathematical) Problems  
April 10th, 2018, 04:32 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 19,722 Thanks: 1807 
I think romsek meant that a regular ngon 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  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,446 Thanks: 2499 Math Focus: Mainly analysis and algebra  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:
Last edited by v8archie; April 10th, 2018 at 04:44 AM.  

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