The golden ratio | Introduction to Euclidean geometry | Geometry | Khan Academy

An introduction to one of the most amazing ideas/numbers in mathematics

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Geometry on Khan Academy: We are surrounded by space. And that space contains lots of things. And these things have shapes. In geometry we are concerned with the nature of these shapes, how we define them, and what they teach us about the world at large--from math to architecture to biology to astronomy (and everything in between). Learning geometry is about more than just taking your medicine ("It's good for you!"), it's at the core of everything that exists--including you. Having said all that, some of the specific topics we'll cover include angles, intersecting lines, right triangles, perimeter, area, volume, circles, triangles, quadrilaterals, analytic geometry, and geometric constructions. Wow. That's a lot. To summarize: it's difficult to imagine any area of math that is more widely used than geometry.

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[nonchalantd]

I realize how interesting math can be now that I am not under pressure to learn it.

[jarjarquan]

I hate math at school. It's stressful and I dread doing the work. But when I'm learning about it on my own (like watching this video) I love it!

[undeddjester]

Wow... Got given a programming problem to solve looking at fibonacci's sequence, read the equation and could not see where the hell it came from. Spotting something about "The golden ratio", I decided to go searching for that, which in turn led me here...

How the hell can such a simple idea of having a line with the ratio of "a to b" being the same as "a+b to a" produce something so fricking awesome!? Mind officially blown. Thanks muchly ;)

[jayveritasmusic]

This is great! I'm frustrated though that math is being dumbed down in public school. Like the math in this video isn't that complex, but when I show it to my teacher in ninth grade HONORS its treated like some kind of disease and I'm being told that it's too "hard" to do in class. Like seriously we should be allowed to learn at our own pace and not be put down to learn simple formulas that we've learned in the sixth grade!

[Adam-jotr]

Sal I love how excited you get about math and concepts like this. I feel like these things can be mind-blowing and it's so cool to have someone to share this with, even if it's just me at home watching your videos. So proud of everything Khan Academy has become

[carlosalexandreFAT]

The association of the main numbers in the field of mathematics with each other, reflects numerical sequences that correspond to the dimensions of the Earth, the Moon, and the Sun in the unit of measurement in meters, which is: 1' (second) / 299792458 m/s (speed of light in a vacuum).

Ramanujan number: 1, 729
Earth's equatorial radius: 6, 378 km.
Golden ratio: 1.61803...
• (1, 729 x 6, 378 x (10^-3)) ^1.61803 x (10^-3) = 3, 474.18
Moon's diameter: 3, 474 km.

Ramanujan number: 1, 729
Speed of light: 299, 792, 458 m/s
Earth's Equatorial Diameter: 12, 756 km. Earth's Equatorial Radius: 6, 378 km.
• (1, 729 x 299, 792, 458) / 12, 756 / 6, 378) = 6, 371
Earth's average radius: 6, 371 km.

The Cubit
The cubit = Pi - phi^2 = 0.5236
Lunar distance: 384, 400 km.
(0.5236 x (10^6) - 384, 400) x 10 = 1, 392, 000
Sun´s diameter: 1, 392, 000 km.

Higgs Boson: 125.35 (GeV)
Golden ratio: 1.61803...
(125.35 x (10^-1) - 1.61803) x (10^3) = 10, 916.97
Circumference of the Moon: 10, 916 km.

Golden ratio: 1.618
Golden Angle: 137.5
Earth's equatorial radius: 6, 378
Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2.
(((1.618 ^137.5) / 6, 378) / 6.67) x (10^-20) = 12, 756.62
Earth’s equatorial diameter: 12, 756 km.

The Euler Number is approximately: 2.71828...
Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2.
Golden ratio: 1.618ɸ
(2.71828 ^ 6.67) x 1.618 x 10 = 12, 756.23
Earth’s equatorial diameter: 12, 756 km.

Planck’s constant: 6.63 × 10-34 m2 kg.
Circumference of the Moon: 10, 916.
Golden ratio: 1.618 ɸ
(((6.63 ^ (10, 916 x 10^-4 )) x 1.618 x (10^3) = 12, 756.82
Earth’s equatorial diameter: 12, 756 km.

Planck's temperature: 1.41679 x 10^32 Kelvin.
Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2.
Speed of Sound: 340.29 m/s
(1.41679 ^ 6.67) x 340.29 - 1 = 3, 474.81
Moon's diameter:: 3, 474 km.

Cosmic microwave background radiation
2.725 kelvins, 160.4 GHz,
Pi: 3.14
Earth's polar radius: 6, 357 km.
((2.725 x 160.4) / 3.14 x (10^4) - (6, 357 x 10^-3) = 1, 392, 000
The diameter of the Sun: 1, 392, 000 km.

Numbers 3, 6 & 9 - Nikola Tesla

One Parsec = 206265 AU = 3.26 light-years = 3.086 × 10^13 km.
The Numbers: 3, 6 and 9

((3^6) x 9) - (3.086 x (10^3)) -1 = 3, 474
The Moon's diameter: 3, 474 km.

Now we will use the diameter of the Moon.
Moon's diameter: 3, 474 km.
(3.474 + 369 + 1) x (10^2) = 384, 400
The term L.D (Lunar Distance) refers to the average distance between the Earth and the Moon, which is 384, 400 km.

Moon's diameter: 3, 474 km.
((3+6+9) x 3 x 6 x 9) - 9 - 3 + 3, 474 = 6, 378
Earth's equatorial radius: 6, 378 km.


By Gustavo Muniz

[TrusterVision]

The Golden ratio is also found the our DNA helix, the bronchial tree structure of the lung, the heart valves, This is not just a ratio... It is nature's ratio. Very intriguing indeed.

[TimothySmithMIPM]

Very straight forward. Just look at the basic definition: φ = a/b.
Thus if a = 10, φ = 10/b = 1.618, or b = 10/φ
And that is even more fun if you remember that 1/φ = φ - 1, so you can do that in your head.
φ - 1 = 0.618 times 10 = 6.18 units (cm for this case)
So side A = 10, side B = 6.18...
GREAT video!!
PS, obviously I'm only using the first 3 digits of φ for calculations.

[avikaalgond]

Very interesting a great wonder i just liked it a lot Thanx for dis wonder creation.!!!

[monzur]

10:10, "Let me scroll down a little bit, " scrolls up.

[alexkeila]

One of the best Khan Academy videos ever!!! I have watched dozens of times!! Every time I think of φ, I return to this video just for fun!!

[LHerne]

you could go this way as well: a^2 -ba -b^2 = 0 and solve for a, choosing the positive answer: a= b(1+sqrt(5))/2. Then use this in a/b and get the golden ratio as (1+sqrt(5))/2

[Risayume]

I personally found this video very confusing. It didn't really explain what phi was in the beginning of the video and went right into some equation that was afterwards explained as the golden ratio. I was beyond confused. I also would like to see that the work is more organized, it looks like its written it all over the place.

[mdhussain]

Such a beautiful equation I ever see in my life. I love Mathematics
😢😢😢

[fredlunau]

Started watching this as a goof to myself, and just couldn't stop watching. Now THAT is teaching.

[Frisbinator]

I read 1/2 of the book about the golden ratio by Mario Livio, I gave up on it because I wasn't really grasping all that was said, but this video really cleared things up, I'll probably head back into it now, thanks so much for your time and effort, the way you use colors and simple explanations is very helpful and clear. 

[mohammads]

I followed everything it was so awesome. I was doing a research essay on this topic and I got an A+ on it.

[OmicronIII]

Having never studied them myself, I have to take Discovery Channel's word for it that the Golden Ratio was used extensively in ancient architecture; the Parthenon being one of them. Also as I understand it, there are a great number of archways with this ratio measuring from base to sides and up to just where it terminates with the arch its self. It’s also in the windows of some churches. (as I understand it)

[LiwaySaGu]

This video is the best fundamental mathematical explanation of the golden ratio online. THANKS a

[Ifrumnoj]

I have always loved Math. I do stem talks at my kids schools and I tell them math is the key to the universe which it is.