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October 23rd, 2012, 01:39 PM  #1 
Senior Member Joined: Jun 2011 Posts: 164 Thanks: 0  Prove the area formula of a circle?
Someone asked me this question: Given that there is a circle with radius r, prove that the area of the circle is pi * r^2 given that you know the circumference of a circle is 2pi*r and there are 2pi radians in one revolution. He asked me to do it using limits. I knew how to prove that using definite integral, but he says he wants me to do it using limits, and there are two ways to do it. I got stuck and come here for help I am not very good at Calculus yet, if there you could explain how you got to your answer when you tell me it, it would be great! 
October 23rd, 2012, 03:30 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Prove the area formula of a circle?
The area of an ngon is: hence: 
October 23rd, 2012, 03:36 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond  Re: Prove the area formula of a circle?
You can divide a circle into several small triangles having two sides equal to the radius, r, and having angle ? opposite the arc of the circle. It's easier to deal with right angled triangles, so we bisect ? with a line down to the base of the triangle. The base is the segment between the two radii. There are 4?/? of these right angled triangles. Their area is So our limit is which can be written as I'm assuming you know 
October 23rd, 2012, 03:43 PM  #4 
Math Team Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions  Re: Prove the area formula of a circle?
Actually when using integration you are using limit. 
October 23rd, 2012, 05:50 PM  #5 
Senior Member Joined: Jun 2011 Posts: 164 Thanks: 0  Re: Prove the area formula of a circle?
Thanks guys! That was very helpful, I did not realize circle could be cut into infinitely amount of pieces, which was probably why I struggled with it As for the integral, I don't think he ever wants to see an integral in this problem, so.... 

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