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!!

## Comments

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TopNewestGreat! Too bad we can't do something like this for ellipses! – Michael Mendrin · 2 years, 6 months ago

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– Trevor Arashiro · 2 years, 6 months ago

HAHA, If only.Log in to reply

Proof that area of a polygon =ap/2??? – Harsh Shrivastava · 2 years, 6 months ago

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– Trevor Arashiro · 2 years, 6 months ago

posted it as well as another proof. This is turning into a proof chain XD.Log in to reply

@Trevor Arashiro – Harsh Shrivastava · 2 years, 6 months ago

Thanks,good proofLog in to reply

– Trevor Arashiro · 2 years, 6 months ago

Will do, but just later tonight.Log in to reply

I tried it to pentagon and its correct! Write a comment or ask a question... – Artemis Fortu · 2 years, 5 months ago

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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" =.=||| – 敬全 钟 · 2 years, 6 months ago

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– Trevor Arashiro · 2 years, 6 months ago

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.Log in to reply

We can also prove by nth root of unity lying on circle,finding the area then limit n tending to infinite – Krishna Sharma · 2 years, 6 months ago

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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! – Trevor Arashiro · 2 years, 6 months ago

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