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October 23rd, 2012, 01:39 PM   #1
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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!
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October 23rd, 2012, 03:30 PM   #2
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Re: Prove the area formula of a circle?

The area of an n-gon is:

hence:

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October 23rd, 2012, 03:36 PM   #3
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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
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October 23rd, 2012, 03:43 PM   #4
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Re: Prove the area formula of a circle?

Actually when using integration you are using limit.

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October 23rd, 2012, 05:50 PM   #5
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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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