SOLUTION: what is an indirect proof of Theorem 11.9 (area of a circle)? *the area of a circle is pi times the square of the radius.

Question 409846: what is an indirect proof of Theorem 11.9 (area of a circle)?
*the area of a circle is pi times the square of the radius.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
The indirect way would be to assume that the area is not equal to and find a contradiction. However this would involve finding the actual area of a circle, so an indirect proof is unnecessary.

Instead, construct a regular n-gon with n isosceles triangles (similar to "Theorem 11.8 which I recently solved, you can find it under the solutions I posted). Instead of finding the length of the base, we find the area, and multiply by n.

Recall that the area of the triangle is where . We have n of these triangles, so the area of the n-gon is

(this may or may not be a good formula to memorize, but it's more important to know where it came from). Take the limit as n goes to infinity:

. Just like the last problem, if we let ,

. Applying L'Hopital's rule,

. Since (by direct substitution and noting we obtain a finite and determinate number),

, hence the area of a circle.

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