Why does π appear in circle area?

Explains why the area of a circle is exactly equal to π R^2

[Mike-nfnf]

My main problem with math in physics isn't so much understanding how to go through the equation, but rather, WHY that particular equation is the correct one. These videos help explain why. The color-coding and models are very helpful in making sense of the information. Thanks for all the years of work on your channel!!

[bo-dgbh]

when a physics guy does mathematical proof, it feels like he’s making sense of things instead of simply stacking up equations

[azathoth]

Love that you can do incredible videos on how tensors work and then do some beautiful proofs on elementary geometry

[VlanimationTales]

Beautiful explanation! I know a few proofs for the area of a circle, but I've never seen it presented this way.

BTW, congrats on one million subscribers! 🎉

[KVII]

such a beautiful derivation of a very common equation in math and physics, thx!

[e.b.]

But where does pi come from in the circumference?
I took those triangle areas and came up with the limit
Lim n-->inf (n)(r*sin(pi/n))(r*cos(pi/n))
x=pi/n
Lim x-->0 (pi/x)(r^2sin(x)cos(x))
[Lim x-->0 pi*r^2][Lim x-->0 sin(x)/x][Lim x-->0 cos(x)]
(Pi*r^2)(1)(1)
Pi*r^2
But the pi still comes out of nowhere lol

[KaliFissure]

What's cute is that if you have a circle with radius Pi, the circumference is 2Pi² and the area is Pi³. The area of a sphere of radius Pi is 4Pi³ and the volume 4/3 Pi⁴

[tseckwr]

thank you again for your hard work on these excellent videos you produce.

[VADemon]

As that other comment asking about where π comes from in the circumference, I expected some form of introduction through the ways the ancient Greeks might've derived π.

[Lumpi_]

The music is just as awesome as the content!

[feynstein]

This seems a bit circular (heh). So in essence, this happens because the area of a circle is R/2 times bigger than its circumference.

[yodaqwq]

But how do we know that the ratio of the circumference and diameter is the same for all circles?

[roaster]

I use diameter squared times .785 for area of a circle, on the job related to pipe flow rate

[alihakimtaskran]

Eugene used to share videos on complex topics like quantum mechanics and general relativity. Could you share more about those complex topics?

[erdaly]

Eugene. you are a role model! Thank you so much

[Rationalific]

It's an amazing look to see why things are the way they are, rather than just taking them for granted as the way they are. The explanation was amazing. As for my brain, I could barely keep up, and this isn't rocket science. My mind boggles at those like the creator of this video who can not only easily understand but also explain these ideas so well. And then there are those who actually figured these things out....the giants upon which we and all of modern society stands (mostly aloofly and ungratefully).

[LawatheMEid]

C=π x diameter
This equation come from arc length equation which is:
A = angle of the arc in radian x radius
So:
C = 2π x r

[FizykaFilozofiaFuturystyka]

Elegant proof :D
Coincidentally I was wondering when will You upload Your next full-length video, and I am happy You did :D but its too short for me to be fully satisfied, but it's better than nothing! :D I love Your channel and hope for more videos

[antoniomoyal]

The proof that mathematical objects cannot be material. As matrer cannot be divided infinitely.

[pankajtomar]

Excellent vedio. Nice explained. I got interest in physics due to your channel in my age of 50 .
Regards