Area of a regular polygon

In this video, I find the area of a regular n-gon (a polygon with n equal sides), and show that as n goes to infinity, the n-gon converges to a circle! So in this sense, a circle is an infinity-gon, how cool is that?

[conrad]

"N-Gon, got to integrate them all." 😀

[chonchjohnch]

Once back in high school, I tried figuring out the areas of regular polygons with side lengths of one just using the number of sides. Got in trouble for not paying attention 😂😂

[Jaylooker]

This is a useful construction to consider if the regular n-gons are taken as finite groups or simplicial complexes and the circle is taken as the fundamental group.

[aadityabhetuwal]

the name of shape sounds like it came fresh from chemistry

[gustavoorocha]

I am Brazilian and studying math. I watch your videos to improve my english and get some math tips.

[marcialvalero]

This video is really cool!, I loved this proof ❤️

[michalbotor]

so for a physicist every polygon is a circle! ^_^

[gedlangosz]

Couple of points:

1) use the formula for area of a triangle taught at school
Area= 1/2 a b sin C
This will save a lot of time

2) introduce a change if variable, p=1/n and then use L'Hopital. You can then show how to get the answer without having to refer back to another video.

[colleen]

So how do we do this? “Simply using calculus”.

[thedoublehelix]

"sin(love)/love ... so sin(junk)/junk". Having some heart troubles Dr. P?

[Austin]

N triangles, using .5absinc you get .5r*r*sin(2pi/n). So total area is N*r^2*sin(2pi/n)/2

[evelynmisaki]

I'm glad I found this channel! It's really good! Though, I found it through some weirdo homophobe whining about being blocked and trying to play victim. Honestly, good on you for blocking such people and not tolerating that kinda shit. It means a lot to LGBTQI people like me. I really love this proof too, and have been watching a fair few of your videos today, subbed! 💗

[weerman]

Can you prove the formula for the area of a circle using determinants?
Because you also made a video where you found the area of an arbitrary polygon with determinants.

[TheMauror]

Very nice proof Dr peyam! What about the perimeter? Will it converge too?

[dougr.]

At about 9:00 I already saw that using sine theta is approximately (and more accurately, is equal in the limit to) theta as n goes to infinity and so pi r squared is obtained

[HawluchaMCPE]

Interesting question, I'll try the question myself before watching this:)

[kpp_]

Where did you get Pi on 9:25? You could have introduced any constant != inf or != 0 and got a C*r^2.

[tmlen]

with that formula a 1.5-gon would get a negative area

[gustavoorocha]

could you point me to a basic math book in English?

[lamigodiomedess.]

Please help. I cant find.. Who discovered area of regular polygon?