Spherical Geometry: Deriving The Formula For The Area Of A Spherical Triangle

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Summary:
► A spherical triangle is a surface area of a sphere bounded by 3 arcs of great circles.
► Any spherical triangle has it's antipodal duplicate
► Each of these spherical triangles are intersections of 3 different spherical lunes.
► Adding up the area of all 6 lunes results in the surface area of a sphere with 4 additional spherical triangles.
►Girard's Theorem: the area (T) of a spherical triangle, with interior angles a,b and c, is given by T = r^2 (a + b + c - pi). The
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Music promoted by Audio Library • Chill - sakura Hz (No ...

[@MuPrimeMath]

5:05 Flex on your friends by telling them that beach balls are actually colored in 6 spherical lunes

[@michaeltedders1650]

mathematically,Its called hosohedron

[@vladanikin696]

You can flex even more by calling those lunes spherical digons. That's an actual name, no joke

[@paulfoss5385]

Vlad Anikin Never understood why they used the prefix di instead of bi which would be easier for people approaching the subject. I think it is long past time we just let digons be bigons.

[@yimoawanardo]

@@paulfoss5385 story of having the Greeks understand you too lol

[@vladanikin696]

@@paulfoss5385 because of consistency. Di- is from Greek and bi- is from Latin so using bi- instead of di- in digon would also make heptagons into septagons. In the end it doesn't matter if you understand how this prefixes work.

[@jonipaliares5475]

Awesome video! The formula at the end also shows that the angles inside a spherical triangle will always add up to more than π or 180°.

[@jonipaliares5475]

@@cubing7276 I'm not sure if that's what you meant but, on planar euclidean triangles the angles add up to exactly π and for hyperbolic triangles they add up to less then π.

[@dougr.2398]

Good insight

[@calyodelphi124]

The most important trick that goes unstated in this, that took me a good ten minutes to figure out, is how the addition of all six lunes gives you the surface area of the sphere plus four triangles: 1. When you add the two alpha lunes, you get just the area of those lunes. 2. When you add the two beta lunes, the areas of those lunes overlap with the areas of the alpha lunes where the triangles are located. So you add the areas of the beta lunes PLUS two areas of the triangle. 3. Same happens with the gamma lunes. You are adding the areas of the two lunes PLUS two areas of the triangles because of overlap. And there you have it. Adding the six overlapping lunes gives you the surface area of the sphere plus four triangles. Just substitute α,β,γ into L(θ)=2θr², do a little algebra, and you get r²(α+β+γ-π)=T

[@a13xcube5]

Thank you so much!

[@pvasudev1967]

I would like to add that the area of alpha lunes also includes the area of two spherical triangles. The area of two triangles has to be added once but is added three times -six times the area of the spherical triangle. So four times the area of the triangle has to be subtracted from the area of lunes to ensure that the area is added only once in order to calculate the area of sphere from the total area of the lunes.

[@aaaaaa-rr8xm]

basicaly the 4 is 3*2-2 but I dont understand why its 2pi and not 180

[@molybd3num823]

@@aaaaaa-rr8xm radians

[@esterina7772]

life saver.

[@stockimage1437]

Can’t find this quality anywhere else on KZbin. Keep up the good work 👍

[@ThinkTwiceLtu]

Thanks for watching :)

[@theunknown4834]

3blue1brown

[@randomdude9135]

H1N1

[@mastershooter64]

gotta say it this guy is pretty good, but 3blue1brown is equally as good if not more so!

[@leif1075]

@@ThinkTwiceLtu What would an angle of zero degrees between two great circles look like then..the same as an angle of 2pi or 360 degrees?..as far as i can tell it would..since either way the circles overlap entirely..

[@mathemaniac]

Great video! I didn't know this before! By the way, in these days, no matter where you are, stay healthy.

[@ThinkTwiceLtu]

Thank you:)

[@samuelgantner4609]

I was just wondering if someone has stolen Trevor Cheung's Quora profile-pic but then I realized that this is actually you. :)

[@richardwolfendale8783]

This channel is way to underrated, it's much better than other maths channels with millions of subscribers

[@hamiltonianpathondodecahed5236]

-presh talwalker is the first name that comes to the mind-

[@That_One_Guy...]

*Fresh Toadwalker

[@That_One_Guy...]

Other than him, there's many good math channel with millon of subscriber that you can try checking out first

[@rasoulkhoshravan5912]

I am sharing this video on my facebook page in order to promote it. Hope this will help.

[@duckymomo7935]

A lot of math channels that really don’t deserve the recognition sure

[@chrisj7903]

More great geometry. You've a great knack for leaving each scene up for just the right amount of time for my brain to compute what you just emphasised, and that makes all the difference.

[@mrkhunt.]

Great video, that “Aha” moment was at 4:55 to add up and find the equation. 👏👏👏

[@ThinkTwiceLtu]

Thanks!

[@Aditya-khurmi]

Wow, your video quality improved a lot! Feeling so good for you.

[@ThinkTwiceLtu]

Thanks:)

[@rutvikpanchal466]

Can you make a playlist of all of your videos so i can listen to the amazing selection of the songs you use, pls. Edit : I'm not saying that i just wanna listen to songs, i love all of your videos, but your selection of songs is just perfect.

[@MrPinknumber]

I love these videos, it makes it so much easier to understand. Great work, keep it up :)

[@mr_rede_de_stone916]

All about this was so desperately beautiful, from the formula itself to the yet quite simple proof and the animation - this was both highly interesting and soothing, with a great timing in the video, pausing long enough to ensure understanding but not to much to keep it fluid. Love it!

[@ThinkTwiceLtu]

Thanks a lot:)

[@AntoCharles]

Good 👏 Quarantine👏 Content👏 All jokes aside awesome job :D

[@ThinkTwiceLtu]

Thanks 😊

[@John_does]

Interesting conclusion from that, which is a bit trivial in highnsight, is that every triangle and in extension every polygon on a sphere with non zero area, the sum of it angels is necessary bigger then that kind of polygon on the plane, does this extend to higher dimensions with oclidian spcaes?

[@sayamqazi]

euclidian *

[@tsawy6]

Ahhh, and on a plane, alpha+beta+gamma would go to pi, making the term inside the brackets go to 0, bit simultaneously, r would approach Infinity!

[@ArturHolanda91]

Well pointed

[@brandonklein1]

Well, this reflects that if we place a planar triangle on a sphere, only 1 point touches the sphere; so we have no area.

[@amaarquadri]

Beautiful proof for a beautiful formula!

[@snillor999]

I cannot believe how much this makes sense.. Thanks for the awesome graphics and the intuitive explanation. Keep up the great work.

[@ThinkTwiceLtu]

Thanks, happy to hear that!

[@gbporto9701]

That was beautiful!! Such a high quality content, thank you!

[@ThinkTwiceLtu]

Happy to hear that:)

[@andreyzyablikov9891]

I do not speak English (greetings from Russia), if only a little bit, but your videos are understandable, because you speak immediately in 3 truly international languages, these are Music, Beautiful and Clear Visualization and Mathematics! Thank you very much for your wonderful videos!

[@ThinkTwiceLtu]

Спасибо)

[@nagys36snn]

real nice video, nice, smooth transitions, clear, decluttered, followable, i loved it!

[@ThinkTwiceLtu]

Thanks 😊

[@EdwardNavu]

I expected Flat earthers being pissed here, then I realized that Flat earther won't bother to seek knowledge, so they won't be here.

[@IsraelIsLikeWater]

How would this be useful in proving we inhabit a spherical earth? To actually measure spherical lunes (of they were to exist) wouldn’t be an incredible feat (given the presupposed size/shape of the earth). People could try to get a smaller portion, but they’d be basing it on a presupposition of the radius- their confirmation bias.

[@Binyamin.Tsadik]

Beautifully done, love the animations. Keep it up man!

[@ThinkTwiceLtu]

Thanks!

[@girishgarg2816]

Having to get sit at home and watch your videos. Pure bliss!

[@brucea9871]

Very interesting video. This is something we never see in high school - formulas in spherical geometry (much less their proof). In fact geometry seems to be somewhat neglected in school in favour of algebra (which may appear to be more pratical and is preparation for calculus). Sure we learned basic facts like the area of a triangle is half the base times the height but in school I always wondered what if we do not know the height? What if we only know the lengths of the three sides (which is much more likely)? It was only many years after high school I discovered Heron's formula; the area of a triangle is the square root of s(s-a)(s-b)(s-c) where a, b, and c are the lengths of the three sides and s = (a + b + c)/2. There are many other fascinating facts about objects as seemingly simple as triangles, such as every triangle has both an incircle (a circle tangent to each side) and a circumcircle (a circle which passes through each vertex). This is not true for polygons in general. Also each of the following three sets of segments in a triangle intersect in a point: 1) the altitudes; 2) the medians; 3) the angle bisectors; 4) the perpendicular bisectors. Then there is Ceva's theorem which shows a connection between the the lengths of the segments cut off on each side by three cevians (a segment drawn from a vertex to an opposite side) passing through a single point. And this is just triangles; there are many other fascinating facts about polygons and geometry, spherical geometry, and three dimensional geometry. Another topic neglected in high school is number theory. There are many fascinating facts about the integers and certain sequences of integers (such as the Fibonacci sequence) and some very interesting but unsolved problems (such as whether there are an infinite number of twin primes and the Collatz conjecture).

[@ronaldmorris8390]

So well done! Kudos. Keep up the great work!

[@sasmitarath4312]

Incredible. You are maths' makeup artist

[@emanuellopez8578]

I usually don't watch your vids if I don't know what you're exposing, but I'm gonna watch it all now bc you deserve all the attention you receive and so much more, thanks for this I really appreciate it

[@ThinkTwiceLtu]

Thanks for the support:)

[@samaryadav7212]

I was trying to visualize this from 2 days but the animations awesome I got that at once. THANKS

[@farisakmal2722]

Your videos are pure eyegasm

[@jeedux5804]

A very nice presentation on the area of spherical triangle. I learned a lot. Tnx

[@OCEAN_NINJA]

I can't think something like this , this so beautiful visualization

[@NonTwinBrothers]

Very nicely animated!

[@jacabezaa_4026]

Great! Now I know how to derive the formula to the area of a spherical triangle!

[@ThinkTwiceLtu]

:)

[@MegaBubbles360]

i didn't know how much i needed this video. so pure 🥺

[@ThinkTwiceLtu]

:')

[@DiegoMathemagician]

Pretty cool, I thought that in order to understand the proof of this I was going to need very sophisticated mathematics. I am really grateful because you taught me a lot of hidden gems in geometry.

[@jiaming5269]

Now, THIS. THIS IS IT.

[@yeast4529]

These really are some of the most amazing videos on KZbin

[@ThinkTwiceLtu]

Thank you:)

[@joni5028]

Great Video! Keep up the good work. Wonderful proof thanks to the animations

[@ThinkTwiceLtu]

Thanks:)

[@avyakthaachar2.718]

Awesome animation! Thanks a lot 🙏

[@eliyasne9695]

This is beautiful! The proof i knew to this formula uses the fact that the area is proportional to the total curveture inside the spherical triangle.

[@roy04]

It took me less time to understand areas of triangles in spherical geometry from this video than I the time I took to understand why Heron's formula (basic euclidean geometry) works from actual math classes

[@lopkobor6916]

This proof is so beautiful and simple resulting in an elegant formula that's also beautiful and simple.

[@vma011]

I'm drooling myself. This is beautiful. Loved it!

[@ThinkTwiceLtu]

Thank you!!

[@krishism]

Incredible Stuff! I loved the presentation.

[@ThinkTwiceLtu]

Thank you

[@Invalid571]

Your videos are always a joy to watch! 👏 👏 ☺ Keep going, you are an inspiration for all mathematicians/math students! Edit: I especially like the chill lofi music. ☺

[@ThinkTwiceLtu]

Thank you:)

[@catherinebuhat3657]

GREAT VISUALS! SUBSCRIBED

[@Prabhav26]

👏Great Animation👏. Visualization helps a lot. ❤Keep making such quality content ❤

[@user-jn1sv8oq7b]

Why are the sides of triangle only "Great Circle"? Can the sides be any smaller circles?

[@quacking.duck.3243]

It can but it won't be considered a proper triangle anymore, since the only geodesics on a sphere are the great circles. Geodesics are lines which minimize distance, and are defined by the local geometry; in the plane they are straight lines for example.

[@user-yy5ru3jl5b]

That's literally the best video I have ever seen.

[@EmanuelSygal1]

Captivating and Beautiful, thank you!

[@ThinkTwiceLtu]

Thanks for watching!

[@quantumcity6679]

You're work is excellent..... Good job

[@quantumcity6679]

😂......ohh... I didn't see that....thanks for *tailling* me

[@lovemath980]

Great, thanks so much. How duid u do this animation?

[@thebigoeuph]

Very nice video! I am curious though, in the limit of large radius (very little curvature) we should expect that even though the triangle would get very big, does the formula reduce back to good ol bh/2?

[@sohanaiyappa3515]

Technically a straight line can be a considered as an arc of a circle of radius infinity . So in that way, every 2d triangle is a spherical triangle with the sides being arcs of great circles of a sphere with radius infinity

[@cosimobaldi03]

Seriously epic, mate. It's a great channel

[@ThinkTwiceLtu]

Thanks!!

[@pyrokinetikrlz]

Interesting to note that the sum of the internal angles of a spherical triangle is NOT pi radians as in the case of the good old plane triangle. The area of the spherical triangle would be zero, according to the equation obtained in this video.

[@Nickesponja]

Man, these videos are amazing! Could you animate a proof of L'Hopital's rule? I'm sure there has to be a neat visual proof

[@Starrkgamingy]

Galing Naman daming matutunang Tayo dito.. thank you for sharinng

[@gonzalochristobal]

great video and beautiful animations!

[@ThinkTwiceLtu]

Thank you :,-)

[@keyyyla]

This is amazing. May I know which software you use for your animation?:)

[@ThinkTwiceLtu]

Processing and c4d:)

[@lucasnahp]

Thanks for this helpful video

[@alejrandom6592]

Nice video, thanks for posting 😊

[@gabrielmello3293]

Great quality. Keep it up.

[@ThinkTwiceLtu]

Thanks!

[@josedanielbazanmanzano9607]

This really is one of the most beautiful proof in all maths

[@alpe6127]

Beautiful video as usual, thanks for uploading! I have a question however, the formula for the Area is: T = r² (a + b + c - pi) The sum of the angles in a triangle equals 180° = pi, so: a + b + c = pi Doesn't that mean that the Area is always 0. (Is it because the sum of the angles of a spherical triangle does not equal pi?)

[@Spieder02]

Yes imagine walking from the equator to the north pole turn 90 degrees and then go down to the equator turn 90 degrees again and go back to the point where you started that triangle has 270 degrees you can have many different numbers bigger or equal to 180 on a sphere

[@alpe6127]

Thanks!

[@paultikotin]

As others have pointed out, this works because the angles of the spherical triangle sum to more than pi. This means we can do an experiment... Set up a very large triangle where the vertices can all be seen and are visible to an observer located at any vertex. Measure the angles. If the universe is Euclidean, you will always get pi. If space is curved you will only get pi in special cases.

[@DamaKubu]

Dot comes to existence, dot moves -> line, line moves -> circle, circle moves -> sphere. *happiness noises*

[@txikitofandango]

This is nice. But how many degrees of freedom are there among the three angles? Does the third one depend on the other two? Hmm maybe not

[@junemotour6888]

Very nice content like it.keep it up

[@agrajyadav2951]

Bruv uplooaaaaddd dudeeee

[@dylanparker130]

beautiful!

[@GammaDigamma]

Excellent you saved my neck thanks...

[@marmikpatel9261]

Superb quality great video. Top math visualisation channel on KZbin.

[@ThinkTwiceLtu]

Thanks for watching 😊

[@lamaspacos]

The proof come so smooth ❤

[@anushrao882]

Excellent video as always.

[@ThinkTwiceLtu]

Thank you!!

[@catapillie]

the final formula is... beautiful!

[@rajatchopra5829]

Wonderful as always👌

[@ThinkTwiceLtu]

Thanks!

[@resadavidchannel]

Very cool!!

[@funkysagancat3295]

OMG THAT WAS FUCKING AMAZING PLEASE KEEP MAKING VIDEOS

[@alwysrite]

just like 3B1B you never dissapoint.

[@ThinkTwiceLtu]

:))

[@MathHacker42]

Beautiful explanation

[@ThinkTwiceLtu]

Thank you:)

[@hectorricardodelacruzmonte2566]

That's considering three great circles What does happen when the circles do not pass by the center?? thanks for the video!

[@AJLuna28]

I came into this knowing nothing and feel like I just learned a new language

[@elnurbda]

Спасибо за видео

[@captainsnake8515]

How do you make these? It certainly doesn’t look like manim. Is it done with code or “normal animation”? Either way, keep up the great work!

[@ThinkTwiceLtu]

It depends on what I am animating but usually it's a mixture of two programs: processing (let's you animate with code) and Cinema4D (3D animation software).

[@AmitKumar-xw5gp]

@@ThinkTwiceLtu please make a "how to make" videos using the softwares you use. It will be a big help to all others.

[@ananyapamde4514]

How do you make these animations? Can you tell me how I can also animate these like you? These are beautiful

[@tzambaprama]

yeah! but this assumes that area varies linearly with the angle which we don't now if its true

[@pool7216]

It's beatiful. Which software do you use, please?

[@ThinkTwiceLtu]

Processing and c4d

[@pool7216]

@@ThinkTwiceLtu Thank you.

[@52.yusrilihsanadinatanegar79]

Thanks, i love it

[@sergten]

Great animation!

[@ThinkTwiceLtu]

Thanks 😊

[@Louiscypher93]

Can you do the same for hyperbolic surfaces?

[@jeffbezos3942]

Very impressive

[@tiandao1chouqin]

Brilliant indeed!

[@ebiushardy4078]

Does anyone know what programming language or software is used to create animations like this ?

[@portr002]

Beautiful!

[@ThinkTwiceLtu]

Thanks:)

[@luisgcr]

Beautiful 🤩

[@jwpogue]

A bit complicated, but an amazing video well explained!

[@ThinkTwiceLtu]

Thank you:-)

[@princesoni7241]

Please explain visually the infinite sum of natural numbers. Can you do it .

[@slavikvsvega]

For a flat plane, r is infinite and alpha+beta+gamma-pi=0 so the area can be anything

[@msolec2000]

Angular excess times radius squared. Very nice explanation!

[@ThinkTwiceLtu]

Thank you 😊

[@SOBIESKI_freedom]

How do you get four additional spherical triangle areas (T) when you add up the areas of the six spherical lunes?? I'm stumped at that bit... Could someone explain?

[@ThinkTwiceLtu]

Two of the six triangles are counted towards the surface area of the sphere leaving 4 additional triangles.

[@SOBIESKI_freedom]

@@ThinkTwiceLtu Ah yes. Thank you. 😃 I just needed to ponder on it more deeply than i would normally do.

[@bastiana.n.4277]

Great video!

[@ThinkTwiceLtu]

Thank you 😊