I had the random urge to post a note. So here it is.

The area of any regular polygon can be expressed in the form \(\frac{ap}{2}\) (My proof of this is here)where a is the apothem (or in a circles case its radius) and p is the perimeter.

Because a circle is a polygon with infinitely many sides, this rule applies to circles as well.

The perimeter/circumference of a circle is \(2r\pi\).

Therefore, the area of a circle can be represented as \(\frac{r(2r\pi)}{2}\Rightarrow r^2\pi\).

Boom. Done. fin. El fine. Koniec. Ende. The end.

NOW GO SOLVE SOME GREAT PROBLEMS WITH YOUR NEW FOUND KNOWLEDGE!!

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TopNewestGreat! Too bad we can't do something like this for ellipses!

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HAHA, If only.

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Proof that area of a polygon =ap/2???

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posted it as well as another proof. This is turning into a proof chain XD.

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Thanks,good proof @Trevor Arashiro

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Will do, but just later tonight.

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I tried it to pentagon and its correct! Write a comment or ask a question...

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Thank you for answering the question in my heart that last for 2 years, I wonder why the area must be \(\pi r^2\) and finally I got the answer...In fact, I don't ask my classmates or even teachers (lol) because the answer shall be the same: "This is the rule, it can't be proven" =.=|||

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Well thank you. As I mentioned in one of my other comments, I make these proofs to teach others the simplest and most understandable ways why a formula works. It makes me feel very good to know that someone appreciates what I do. It makes the time that I spend on these notes completely worth it.

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We can also prove by nth root of unity lying on circle,finding the area then limit n tending to infinite

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The reason that I make these simple proofs is because the internet is filled with all these unbelievably complex proofs (such as yours) for the simplest of equations (some of which are necessary, but the majority can be explained in much simpler terms). It's great that they are out there because they are usually very solid and don't rely on "unstable" explanations, such as the fact that a circle is a polygon with infinite sides.

P.S. I definitely don't fall under the category of "smart" people and thus I have no idea what your comment means in the slightest. But it sounds great so I encourage you to post it!

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