Pi Times Phi using a Regular Icosagon Area (visual proof)

In this video, we use a golden triangle to find the area of a regular icosagon sitting inside of the unit circle (the circle with radius 1). From this area formula, we are able to then find an interesting bound on the product of two famous constants: Pi and the golden ratio.

#math​ #manim​ #visualproof​ #mathvideo​ #geometry #mathshorts​ #geometry #mtbos​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #proof​ #iteachmath #icosagon #area #dissection #trigonometry #polygon #sine #pi #goldenratio #phi #goldentriangle #inequality #area

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[prog]

Some can say that we can just plug those constants in a calculator and get the result. But it's not the answer that is beautiful, but the process of solving the problem.

[_Heb_]

Also a pretty neat approximation for 5

[goblin]

No more e^pi vs pi^e
Now it’s Time for phi*pi vs 5

[ИмяФамилия-ери]

special thanks for the lack of background music
I say this as a music lover

[kirahen]

As soon as I saw you calculate the area of one of the triangles of the icosagon, I was like: "OMG I get it that's so cool!" I love these kinds of demonstrations!

[BrunoDerezic]

"I punch these numbers into my calculator, it makes a happy face" - Cave Johnson (portal 2)

[Vengemann]

Your visual concepts are literally the best

[Naman_shukla]

Can you please provide the solution of that Question -> product of chords of an ellipse, you gave that at the last of the video but i am not able to solve it .So please provide the solution ;that video name was-> "product of chords in a circle."

[Froze-rtbk]

Hi u might remember my user on the graphing derivative video that I said I was going to take notes on calculus I actually started to learn it I know how to find limits find derivatives and anti derivatives I'mma bout to take notes on FTC and study the notes

[yurenchu]

5 =
= 10 / 2
= 2.5 * ( 4 ) / 2
= 2.5 * (5 - 1) / 2
= 2.5 * (√5 - 1)(√5 + 1) / 2
= 2.5*(√5 - 1) * [(√5 + 1)/2]
= 2.5*(√5 - 1) * φ
... note: (9/4)² = 81/16 > 80/16 = 5, hence 9/4 > √5 ...
< 2.5*(9/4 - 1) * φ
= 2.5*(5/4) * φ
= (5/2)*(5/4) * φ
= (25/8) * φ
= (3.125) * φ
< π * φ

Therefore,
π * φ > 5


EDIT: This proof is less elegant than the visual geometric proof in the video, because I'm using the knowledge that π = 3.1415... > 3.125 (in other words, my proof relies on having correctly memorised the first few decimals of π, and that's less elegant than simply showing that one area is necessarily greater than another area).

[msolec]

It's cool. I did (355/113)*(233/144) [picked that particular one so that numbers are similar in size. That's 82715/16272.

And since 16272*5 is 81360, which is less than 82715, that means the product is larger than 5... Much less cooler, though

[JoeRussell-ojxm]

Great but at 1:12 you say "we see that the base angle on the blue-shaded triangle is x-1" when you mean the base length

[Ocean-mg]

x/1=1/x-1?

Current convention has it that x/1=x, so x=1/x-1? Isn't that a false statement?

[Nah_Bohdi]

Its pronounced "fi", as in "Phive", and why 5 and Pentagons and Phi are so interrelated.

[rosiefay]

0:08 You call pi pi, so why do you call phi phee? You're talking English, not Greek. And while we're at it, please could you use the familiar symbol ϕ? The symbol you use is confusing.