Here is my much simpler proof for the second part: consider a hexagon inscribed with side length 1. Clearly the circumference of the circle is larger than the perimeter of the hexagon because each 1/6th arc is longer than a hexagon side. So 2pi > 6 so pi>3
@the length of the side of the hexagon is the distance from the centre of the hexagon to a vertex of the hexagon which is the radius of the circle which is 1
@@amansparekh still confused how that proves the line segment joining them tho will be w in the circle? Like ik it is but how's that just clearly true?
@dontspam7186 disc is a convex shape (suffices to prove for unit disc, take two points a, b, the line segment joining them is ta + (1-t)b, ||ta+(1-t)b|| ≤ ||ta||+||(1-t)b|| = t||a|| + (1-t)||b|| ≤ t + 1-t = 1
speaking of π, Im sure you have seen this 'physics' experiment online. Where two blocks and there is a wall. the block nearest to the wall has a constant mass of 1 kg. But if the second block has a mass which is a power of 100. If we move the block with the bigger mass and make it collide with the block with mass 1 kg. assuming no friction and air resistance and all collision has no energy loss. it will travel and collide with the wall again. the total number collisions between the two blocks and the wall corresponds exactly to the digits of π. the channel called 3Blue1Brown explains this in more detail and also better than me
This "proof" scares me, it seems slighttly circular in the usage of radians. If I wanted to be reallly nit-picky, I'd say use purely degrees instead. But even then, I'm not sure how you would define the sine function without using pi.
Hmm perhaps using degrees may have made it clearer. As for the sine function, we only 'needed' to know what sin(30°) is, but this was purely to determine the area of the triangle. We can easily modify this to think of each of the 12 triangles as half of an equilateral triangle and use a more direct approach to calculating the areas.
Don't lie. I studied in Cambridge, and I remember my interview. . No way interview questions are as trivial as this question, which is the kind of thing we did when we were about 12 years old.
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
Here is my much simpler proof for the second part: consider a hexagon inscribed with side length 1. Clearly the circumference of the circle is larger than the perimeter of the hexagon because each 1/6th arc is longer than a hexagon side. So 2pi > 6 so pi>3
Ah nice, I hadn't though of this, that' a nice solution!
I'm confused why's it clear that u can circumscribe a unit hexagon within a circle?
@the length of the side of the hexagon is the distance from the centre of the hexagon to a vertex of the hexagon which is the radius of the circle which is 1
@@amansparekh still confused how that proves the line segment joining them tho will be w in the circle? Like ik it is but how's that just clearly true?
@dontspam7186 disc is a convex shape (suffices to prove for unit disc, take two points a, b, the line segment joining them is ta + (1-t)b, ||ta+(1-t)b|| ≤ ||ta||+||(1-t)b|| = t||a|| + (1-t)||b|| ≤ t + 1-t = 1
👍
Archimedes proved 223/71 < π < 22/7 or 3.1408 < π < 3.1429 by taking 96 - sided regular polygon inscribing and circumscribing a circle (around 250 BC)
also 3.14 < π < 22/7 < √2+√3
speaking of π, Im sure you have seen this 'physics' experiment online. Where two blocks and there is a wall. the block nearest to the wall has a constant mass of 1 kg. But if the second block has a mass which is a power of 100. If we move the block with the bigger mass and make it collide with the block with mass 1 kg. assuming no friction and air resistance and all collision has no energy loss. it will travel and collide with the wall again. the total number collisions between the two blocks and the wall corresponds exactly to the digits of π. the channel called 3Blue1Brown explains this in more detail and also better than me
Yes, I have seen this before - it is remarkable!
Lol i would use pi/4 = 1 - 1/3 + 1/5 - 1/7 ... and group it like this: pi/4 = 1 + (-1/3 + 1/5) +(-1/7 +1/9)... to show that pi/4
This "proof" scares me, it seems slighttly circular in the usage of radians. If I wanted to be reallly nit-picky, I'd say use purely degrees instead. But even then, I'm not sure how you would define the sine function without using pi.
Hmm perhaps using degrees may have made it clearer. As for the sine function, we only 'needed' to know what sin(30°) is, but this was purely to determine the area of the triangle. We can easily modify this to think of each of the 12 triangles as half of an equilateral triangle and use a more direct approach to calculating the areas.
😢
Don't lie. I studied in Cambridge, and I remember my interview. . No way interview questions are as trivial as this question, which is the kind of thing we did when we were about 12 years old.