Mate, you rock. There are some channels that would have taken 15 minutes to explain that, and it still wouldn't have been as easy to understand as your explanation.
@boguslawszostak1784
7 ай бұрын
The area of a flat figure circumscribed around a circle is equal to half of its perimeter multiplied by the radius of the circle. For the right-angled triangle with sides 3, 4, and 5, the area is calculated as 3*4/2=6, and the perimeter is 3+4+5=12. Half of the perimeter is 6, so the radius of the circle is 1.
@user-ry5hm7ho8t
5 ай бұрын
No one says that to me, they just look confused. Maths is great.
@user-ry5hm7ho8t
5 ай бұрын
@@boguslawszostak1784All this stuff is blindingly simple to nerds like us, but it confuses the hell out of some people.
@marklgarcia
5 ай бұрын
@@boguslawszostak1784 Prove it! I would love to see a video on this!
@user-pe9lt3bb6h
2 ай бұрын
some channels would have taken an hour or so! Hahahahhahahahahahhahahahahhahahahahahhahahaha
@JoeTaxpayer
7 ай бұрын
I am 61. The fact that you shared is absolutely remarkable to me, but even more remarkable that I have somehow never run into this before. I work in the math department of a local high school, and I will be sharing this with my coworkers next week. And I will be subscribing to your channel.
@stigcc
7 ай бұрын
What did your coworkers say? Amazing that I never heard this fact before
@circusitch
6 ай бұрын
I am 3.14… old, and this is neat.
@user-ry5hm7ho8t
5 ай бұрын
When they raided Einstien's office after he died, they did not find what they were looking for. Stven Hawkins did not work it out either, but it did concern Pi.
@robdavidson4945
5 ай бұрын
I'm 69 and I find your explanation makes the topic clearer and easy to understand. Thanks for your efforts. I have just subscribed.
@martinkuliza
5 ай бұрын
I'm 50 How is that possible that you've never come across it. i learned this back in high school with 1 difference, we didn't call it a 3 4 5 triangle but other than that the concept is the same but did you , a math teacher , not know this since you are older than me
@tacca2747
6 ай бұрын
What a great explanation. Simply done in such a short time. I'm 73, a retired engineer. It's so good to see a young person with such a special explanatory skill. I've subscribed!
@WTF_.
Жыл бұрын
Amazing Explanation...
@alineharam
7 ай бұрын
I’m 62. Your little explanation made my day.
@sportstalkwithjordan
7 ай бұрын
For any rationally scaled version of a 3-4-5 triangle, the area divided by pi is then rational and a perfect square. Clarification: The area of the inscribed circle is what I mean.
@hcgreier6037
7 ай бұрын
Nice example! I want to point out that the incircle radius of a triangle is always r = A/s, where s is the half perimeter of the triangle and A is the area. This relationship is not so often taught in school. And it don't even need to be a right angled triangle! In this example, we have s = (a+b+c)/2 = (3+4+5)/2 = 12/2 = 6. Area is 3·4/2 = 6. Therefore we get r = A/s = 6/6 = 1, and πr² = π.
@incrediblesam9035
5 ай бұрын
Man thanks for this!
@Masterclass_Geometry
5 ай бұрын
nice 👌
@rickniles6056
4 ай бұрын
Interesting, but calling the perimeter of a triangle (or any not circular shape) it's "circumference" is not standard terminology at least for myself so that was confusing.
@hcgreier6037
4 ай бұрын
@@rickniles6056 Yes, you are right! Sorry for that misnomer!
@quokka_yt
3 ай бұрын
Yes that's exactly how I learned it at my school. We have a lot of exercises using this formula (although we write it as S = pr, where S is the surface area, lowercase p is the semiperimeter, and lowercase r is the inradius)
@tahititoutou3802
6 ай бұрын
Brilliant! Straight to the point in no time!
@gregtitus2467
6 ай бұрын
That was so much fun I ended up with a big grin on my face! I love the 3-4-5 triangle, and never knew this added feature of the a circle inside, and its area. Well done!
@joelbenegh222
3 ай бұрын
And what's the area of the circle OUTSIDE?
@ingiford175
7 ай бұрын
If you know the generators of the triangle, you can find the radius of the incircle. it is simply (a+b-c)/2, so (3+4-5) / 2 =1 So the 5, 12, 13, which gets you 2 as a radius And finally the last of the regular 3 you see on tests; 8, 15, 17: has a 3 radius And yes you can prove all the radius of right triangles with integer values will always be a integer radius.
@clickrick
7 ай бұрын
My mind had gone to exactly this question, and I was about to do the maths. Thank you for getting there first for me!
@ingiford175
7 ай бұрын
@@clickrick This only works for right triangles. If you take a triangle such as a 4,4,4 which is an equilateral triangle with integer coefficients you will not get the same results.
@b1j
7 ай бұрын
Mind blown.
@Lord_Volkner
7 ай бұрын
I'd be curious to see that proof.
@alphalunamare
7 ай бұрын
@@Lord_Volkner Given your other post about squaring the circle I very much doubt that you would understand the proof. Anyway, in the video above replace 3,4 and 5 by x,y and z and out will pop your proof.
@tonylarge5298
4 ай бұрын
On the ranch,back in the 70's a brilliant mathematician showed me this triangle when I was building fences. thank you, I have now learned the mathematics of it. I always made straight fences, rooms, etc. I made a wooden tool of this dimension 50 y ago and passed it on to my grandson.
@bertberw8653
8 ай бұрын
By far the best math videos on youtube
@jfposada007
6 ай бұрын
Brilliant problem! You are very good at explaining it and working through it as well. Bravo, sir!
@josepherhardt164
5 ай бұрын
In fact, any Pythagorean triple (or multiples thereof) will have an inscribed circle with an integer radius. Cf. 5-12-13 or 7-24-25, e.g. More generally, if sides a, b and c are rational, the inscribed circle's radius will also be rational.
@RAFAELSILVA-by6dy
6 ай бұрын
The general form of a Pythagorean triple has sides of length k(m^2 - n^2), k(2mn), k(m^2 + n^2), for m > n > 0 and any k > 0. The radius of the circle inscribed in such a triangle is r = (m-n)nk. Which is clearly a whole number for any Pythagorean triple.
@any2xml
6 ай бұрын
How would you apply this generic for 3,4,5 triangle? I'm not challenging but trying to understand this notation
@thomasanderson9383
6 ай бұрын
@@any2xml set k = 1, m = 2, n = 1. Then the sides of the triangle are 1(2^2 - 1^1) = 3, 1(2*2*1) = 4, 1(2^2 + 1^2) = 5 and the radius is r = (2 - 1)*1*1 = 1
@fullc0de
6 ай бұрын
I think your enthusiasm for math really comes through in this video and is part of what makes your explanation so good.
@billshiff2060
5 ай бұрын
I LOVE this! I am a big fan of Archimedes constant, PI and I am always interested in all the unexpected places it crops up.
@craigcorson3036
6 ай бұрын
That's the most amazing thing that I've learned in a good many years!! I can't believe that I never heard this before.
@CaioTrinchinato
Жыл бұрын
It's amazing that with all the geometry "tricks" still (and will forever be) impossible to square a circle!
@Lord_Volkner
7 ай бұрын
Technically, you can square a circle: The formula for a circle is [x^2 + y^2 = r^2] thus [x^4 + 2(x^2)(y^2) + y^4 = r^4] is a circle squared.
@areadenial2343
5 ай бұрын
@@Lord_VolknerI know this is a joke but it pisses me off lmao
@wcsxwcsx
6 ай бұрын
The really interesting thing isn't that the area is pi, but that the radius is 1.
@timjackson7538
3 ай бұрын
That's what I came here to write. Always worth checking the top comments first 👌🏻
@shampadutta7322
5 ай бұрын
That was a brilliant observation. I've never noticed such a beauty in the 3-4-5 triangle. What's even more fascinating is that the area of the incircle is π = 3.14159... *Notice how the numbers* _3, 4, 5_ *pop up in π, one after another.* And what are the dimensions of the triangle? _3, 4, 5_ . Majestic! *_• Constructive Criticism_* This method was a unique one, using properties of tangents. Another method will be to join the incentre with the 3 vertices. This will divide the triangle in 3 smaller triangles. Now, the 3 sides of the triangle are tangents to the incircle. Hence, the 3 inradii at the 3 points of contacts will be perpendicular to the 3 sides. You can then use the formula of area of a triangle on the 3 smaller triangles and add them up and equate the sum to the area of the larger triangle. This will actually derive you a shortcut: Δ = r · s (where Δ = area of triangle, r = inradius, s = semi-perimeter of triangle) _# Great approach and observation nonetheless._
@hair2050
5 ай бұрын
I’m 56, a bricklayer by trade. Yep, I grew up in an olden days. We walked onsite with a basic plan and very often set out a house foundation and the old 3-4-5 was never left out. We knew no other way, certainly had no other tools at our disposal to do it any other way, and why would we?🤷♂️. It was always satisfying to be the one(s) to set in concrete, brick, 🧱, the building to be.
@SH1974
5 ай бұрын
In farming, for palning new acres (or in my case new wineyards) we used GPS (of course) but we also learned the oldscholl way, using the "12-knot-string" (3+4+5 is 12). Very handy tool! As long as the distances between all knots is equal, it doesn't matter how long the rope or the distance between the knots really is. Using an about 20m long rope gives about 1,5m distance between the knots, its accuracy was astonnishing!
@75blackviking
4 ай бұрын
Dude, brilliant! I'm one of those guys you described who squares rooms with 3-4-5 triangles. Used 'em for 30 years.Had no idea.
@hippophile
5 ай бұрын
Very neat! I have a doctorate in maths and have never seen this (that I recall). It gets better. The other two most common small triangles with Pythagorean triples are 5, 12, 13 and 8, 15, 17. As the "r" in those cases is 2 and 3, the inscribed circle areas are, respectively, 4π and 9π. A general equation for a triple (a,b,c) is easy enough to find (exercise for the reader!). The circumscribed circle is sadly rather dull.
@MathematicFanatic
Жыл бұрын
That is so awesome never knew that! I love 3 4 5 triangles even more!
@goncalofreitas2094
5 ай бұрын
IMO the explanation is even more beautiful and elegant than the result itself, very well done 👏👏
@prakashchandradosi6115
6 ай бұрын
You are always a wanted in my family , today and for ever , no one teaches math as you do 🎉
@brucefoote540
6 ай бұрын
Andy, you dun it! Amazing! and TRUE! Like magic.
@ascendiG
7 ай бұрын
Hi Andy, you have presented a well-illustrated article that displays a result not well known (judging from the comments). It is worth noting, though, that for any integer r, a Pythagorean triangle can be constructed with sides of integral length and having an inscribed circle of radius r with area r2. Obviously, this can be done, trivially, by taking all multiples of 3,4,5. However it can also be achieved with combinations of sides, many of which are relatively prime, as follows. For odd r: side lengths of r2 + 2r, 2r + 2, r2 + 2r + 2. The first three in this sequence are: 3, 4, 5; 15, 8, 17 and 35, 12, 37. For even r: r2/4 + 2r, 2r + 8, r2/4 + 2r + 8. The first three in this sequence are: 5,12,13; 12,16,20 (a multiple of 3,4,5) and 21, 20, 29. The first formula can also produce side combinations for even values of r, but many more of these are multiples of previous results.
@willyh.r.1216
7 ай бұрын
Pretty cool. Thanks 4 sharing.
@jimwinchester339
7 ай бұрын
Wow - I'd never noticed. And the prove was straightforward, too - nothing fancy at all.
@piman9280
5 ай бұрын
ALL Pythagorean triples (right-angled triangles whose sides have whole number lengths) have inscribed circles whose radii have whole number lengths. This means that the area of such circles will always be a whole multiple of pi. Further, if the triple is (a, b, c), then r = (a + b - c)/2 and r = (ab)/(a + b + c). Equating these expressions for r leads to a^2 + b^2 = c^2 (Pythagoras' Theorem). Of further (possible) interest is that the two expressions for r can be stated as: half the difference between the hypotenuse and the sum of the legs, or the product of the legs divided by the perimeter.
@daniellambert726
5 ай бұрын
I can’t believe that no one in the history of humanity has thought of this. It might be too simple, so no genius has devoted time to it.
@timjnoyce
4 ай бұрын
It's also untrue. Draw it to scale and you will see that he is telling porkies
@sistajoseph
6 ай бұрын
This is fantabulous. It's lovely. It's a big thing, wrapped up in a small parcel.
@rodbrittner6524
4 ай бұрын
Totally agree that this is worthy of note! I had no idea. More evidence that the physics and geometry of our universe is often amazingly simple at its core.
@boguslawszostak1784
7 ай бұрын
The area of a flat figure circumscribed around a circle is equal to half of its perimeter multiplied by the radius of the circle. For the right-angled triangle with sides 3, 4, and 5, the area is calculated as 3*4/2=6, and the perimeter is 3+4+5=12. Half of the perimeter is 6, so the radius of the circle is 1.
@Diam0nd_King
6 ай бұрын
Is it just me or does his room look like the room from fnaf 4?
@kmyc89
7 ай бұрын
That's the most beautiful math trivia since 20 years ago
@diegogamba2601
5 ай бұрын
Like everyone else, I was very surprised by Andy's discovery and his beautiful demonstration. I now see that the same conclusion could have been reached in another way. Around a circle of area π we can construct an infinite number of right triangles with legs A, B not necessarily integers. We could call them π-triangles. Applying to these triangles the same considerations used here by Andy, it is easy to show that their hypotenuse is A+B-2 (so the 3-4-5 triangle is a π-triangle, since 5 = 3+4-2) and that their area is A+B-1 (so the triangle 3-4-5 is a π-triangle, because its area is 6 = 3+4-1). Those two general properties lead to a curious corollary: in π-triangles, the perimeter is twice the area. Again, the triangle 3-4-5 is then a π-triangle, since 3+4+5 = 2x6.
@shreeni
Жыл бұрын
The way u explain 🫡
@greendruid33
7 ай бұрын
I love this kind of stuff in math. Thanks for doing these videos!
@AviatorRider
5 ай бұрын
Thank you Andy ! You've explained the deduction in a very didactic way. Very nice !
@theobolt250
5 ай бұрын
YES! Despite my frustrations where math is involved (just me being bad at it while I want to be good at it) I only have admiration for something like this. It has natural beauty. This also feels like it has deeper meaning.
@ajayakumarsamal8108
4 ай бұрын
Excellent explanation. Thanks. Great in explanation too.
@HolySoliDeoGloria
4 ай бұрын
It's interesting that the area is pi, but that just comes from the somewhat less exciting fact that the radius is 1. As other commenters have alluded to, there's an entire class of triangles for which this is true. One way of generalizing them is that this will be true for every triangle (right or otherwise) whose area is half its perimeter. (The 3-4-5 triangle has area 6 and perimeter 12.)
@TheTubejunky
4 ай бұрын
He basically proves how Pathageriom Theorem works.
@kwzon
4 ай бұрын
Great video dude! Love your explanation of every step and the joy you bring to the exploration! 😊
@ronandmary8471
4 ай бұрын
Not sure if this was already stated in the comments below, but this same proof can be used to show that the area of the inscribed circle is an integer multiple of Pi for all Pythagorean triple triangles. Pretty cool.
@birukb273
6 ай бұрын
That’s so awesome how a unit circle can fit perfectly inside a 3-4-5 triangle, I never knew that. How exciting.
@dennisk5818
6 ай бұрын
Awesome. How you solved for the radius, was easy to understand, even if you had basic algebra. I often use the 3-4-5 rule while squaring up things I'm building. That this rule works, is an understatement. That said, learning that an inscribed circle, tangent to the sides, within this triangle, will equal pi, is a wild thing to learn. The same for the lengths of the tangent lines. Too bad they don't (didn't) teach this in any geometry or trig class I took.
@ricardoguzman5014
5 ай бұрын
There is a nice geometric way to solve. Make 4 copies of the triangle, each one with a different 90 degree rotation, which would put the hypotenuses as such: upper right, upper left, lower right, lower left. Place them together to form a 5x5 square. That would make each side length of 3 share a side with a side length of 4, and arranging this way puts a small square right in the center of the 5x5. Now the area of the 5x5 is 25 sq. units. The total area of the 4 triangles is 24 sq units. So the area of the central square is (25-24) sq. units, or 1 square unit, and being a perfect square, it has a side length of 1, which is the same as the radius of the incircle of the 3,4,5.
@Darisiabgal7573
4 ай бұрын
3-4-5 triangles were a bit of a dead end in trigonometry. So for any triangle, there will always be a scalar modification that has the unit circle tgat fits inside of it. It simply means you can scale triangles. 3-4-5 traingles were developed by the babylonians, see plimpton 322 tablet, in order for the construction of rectlinear structures with few nonirthogonal angles. The gate of Ishtar and the Ramp of the Ziggurat of Ur are examples. The problem with the these triangles is that you cannot build circles with them. The only types of triangles that can be used directly to construct circles are the the isosceles triangles, for the unit circle the angle needs to be an integer when dividing 2pi (or 360). These triangles of the unit have a single unique quality, the chord. 120 - SQRT(3) - Triangle 90 - SQRT(2) - Square 72 - SQRT(1-(golden ratio/2)^2) - Pentagon 60 - 1 - Hexagon 51.4 - 0.87 - Heotagon. The only isoceles right triangle is the 90 -45-45. Which the babylonians solved the chord to 6 digit places and pythagoras cult discovered it was a radical. This is possibly the first chord solves. Any unit circle can fit roughly into any polygon but scale of the triangle must be a function of 1/(bisector of the chord). If you take a equilateral triangle, for instance the and you want to inscribe a circle into the triangle. The inscribed triangle has a chord of SQRT(3). The half chord is SQRT(3/4) and since its a unit circle the SQRT(1/4) = 1/2. This means a unit circle will fit into a triangle of scale factor 2, or sides of 2SQRT(3). This means each side is ~3.5. This means the circle fits much more effectively into an equilateral triangle than a 3-4-5 triangle. More over, what if I wanted a structure with integer sides. The inscribed square has sides chord of SQRT(2). If we wanted to inscribe the unit circle into a square, we need to divide it by the 90° bisector which is SQRT(2)/2. Thus give side lengths of 2. Square has combined side lengths of 8, the 3-4-5 triangle has side lengths of 12. The hexagon of sides 2/SQRT(3) encloses a unit circle. At six sides 2*SQRT(3)^2*2/SQRT(3) = 4 * SQRT(3) ~ 7 units. If we continue this process of fitting as the we increase the sides of a equilateral polygon eventually the length of thr sum of the chords is mathematically indistinguishable from the sum length of chords of the inscribed scalar polygon which is indistinguishable from pi. The real break through in trigonometry is the identification of nearly isoceles "perfect" right triangles. This allowed the determination of SQRT(2) to 5th decimal place (base 60 equivilent thereof). It is from these triangles that pythagoras used to formulate his theorem which basically allowed the creation of chord tables, the predecessors to the sins and cosines. Sin x = 1/2 chord 2x Cos x = bisector of chord of 2x So once you have all chords from 0 to 180, you have sin and cosine for 0 to 90. Thank the greeks for wanting to build circular temples instead of square ones.
@icarus-wings
4 ай бұрын
Sir, this is a Wendy’s.
@derjansan9564
6 ай бұрын
Your videos have exactly the right pace for my math level. I love them.
@thesb2171
Жыл бұрын
The way of your explaining. Is exciting 🥰🥰
@bartmix8994
4 ай бұрын
From a retired math teacher to you, Good Job!
@philippeganty
5 ай бұрын
Taking into account its inscribed circle, I always like to point out that the right triangle 3, 4, 5 is in fact a 1 (radius of the inscribed circle), 2 (its diameter), 3, 4, 5. Note that if we extend this triangle by forming the rectangle of length 4 and width 3, the length of the segment between the 2 centers of the inscribed circles is root of 5.
@cherryjuice9946
6 ай бұрын
Good explanation because it was to the point and didn't waste time trying to be dramatic. I'm surprised I never noticed this relation before.
@toolebukk
7 ай бұрын
Could it be that this works for all pythagorean triples? 🤔 EDIT: Wait, that doesnt make any kind of sense, after I started drawing it out 😂 But maybe the ratio between areas at least are the same? EDIT 2: OK, so far I have worked out that the radius seems to always be a whole number. First I thought that the radius was 1 unit longer for every pythagorean triple, but both the 7,24,25 and the 8,15,17 triangles has an inscribed circle with a radius of 3. The 5,12,13 triangle has an inscribed circle radius 2. Next up is 9,40,41 and sure enough the radius here is 4.
@curtishuang5534
7 ай бұрын
Try generalising the sides of th triangle 😉
@romanbayramuk6632
7 ай бұрын
at least they all contain pi
@colinslant
7 ай бұрын
Let (a,b,c) be a Pythagorean triple. By the method set out in the video, we get: c = a - r + b - r c = a + b - 2r 2r = a + b - c r = (a + b - c)/2 So the area of the inscribed circle is 𝝿 * ((a + b - c)/2)^2. Given there are infinitely many Pythagorean triples, can we prove the area will always be an integer multiple of 𝝿? Will r always be an integer?
@curtishuang5534
7 ай бұрын
@@colinslant Yup, so the last step is to see if (a+b-c)/2 is going to be an integer value, or to prove that (a+b-c) is an even number. Pythagorean triples are always three even numbers or two odds numbers and one even number. Both of these cases will result in an even number, hence we should expect an integer multiple of pi
@colinslant
7 ай бұрын
@@curtishuang5534 But we need a proof of that last statement! A Pythagorean triple can obviously consist of three even numbers, since if you multiply the members of any triple by a power of two the result will be a Pythagorean triple containing three even numbers. Is there a proof the only other possibility is two odds and one even? It's probably in Euclid...
@jesuisjamaiscontent
4 ай бұрын
Nice explaination, thank you. Reminds me of deriving Pythagoras' theorem using a square within a square.
@amherst410
5 ай бұрын
I'm obsessed with Pythagorean Triples and loved this problem! I tried it with 6-8-10, etc and enjoyed looking for a pattern.
@ivmusic7714
5 ай бұрын
The pattern is as follows: if the similarity coefficient is equal to k, then the ratio of the areas is equal to k squared ❗️
@theadamabrams
4 ай бұрын
Any Pythagorean triple a,b,c leads to *r = (a+b-c)/2.* In fact that works for any side lengths for a right triangle, integer or not.
@calvinmasters6159
4 ай бұрын
How did I never know? That's gold.
@MrPaulc222
2 ай бұрын
I like to try doing these for myself before checking videos. Vertices are A B C starting at the top and going clockwise. Circle centre is O Triangle area is 6. Split the main triangle into three triangles each with height r. AOC has area (3/2)r. BOC has area 2r. AOB has area (5/2)r. Total area is 6r and is also 6 (as in (3*4)/2) Therefore, r=1 As pi * r^2 = pi, circle area is pi *1 which = pi. Ha ha. I have now watched the video.I went a totally different route from you and, thankfully, got the same result :)
@bordenf
5 ай бұрын
Okay, Andy, this is good actionable information. Just what I needed. Thank you! I'm a math enthusiast myself. And what I would like to know is what is the area of the other three sections within the triangle. Are all three of the sections irrational numbers? Do they have any magical properties when compared to one another? I am dying to know.
@ajayaiyar1907
5 ай бұрын
Beautiful explanation. Learned something new.
@gustavorimoli5585
7 ай бұрын
Very nice, like all the videos in this channel! But I see this more as a riddle than a math question. This is based on the property of the inscribed circle's radius to be equal to 1/3, 1/4 and 1/5 of 3-4-5 triangle sides. Whenever you consider a circle to has a one-unit radius, you will always have a circle with an area equal to pi. In such case, the "trick" is naming such triangle as 3-4-5. On the other hand, if you name it a 6-8-10 triangle (technically also a 3-4-5 one), the resolution won't fit to pi. It's a good one!
@letlhogonolokebasitile6924
7 ай бұрын
Why wouldn’t it?
@Cage66666
7 ай бұрын
It would be double pi since the radius of the circle would be 2 instead of 1 when we double the lengths by making them 6 8 10 instead of 3 4 5@@letlhogonolokebasitile6924 Edit: I meant square of pi instead of double
@Jemmysponz1
7 ай бұрын
it would be 4 pi, no?
@TravisMcGee151
7 ай бұрын
I miss this part in high school 🤔
@tortinwall
7 ай бұрын
I can’t really see what point you’re making. It doesn’t matter what length you call the sides. It’s only the ratios that count. For any 3-4-5 triangle the ratio of the area of the circle to the area of the triangle will be pi/6, and that’s what counts.
@disoldman6573
5 ай бұрын
As a journeyman cabinetmaker I always calculate the hypotheses of a right angle. The 3.4.5. rule can be less accurate.
@bjorncedervall5291
5 ай бұрын
I liked math since before I started school (I am 71 now). Worked a lot with math - never saw this before. Elegant and easy to understand. The only thing that disturbs me is my perception - to me the area of the circle "looks like" it smaller than the rest of the triangle (the area of the triangle is obviously 6 so the area of the circle is slightly more than half of the rest). This must be due to a delusion or misinterpretation that I suffer from. Wonder if anyone else got this feeling. I trust math more than my perception which in this case seems to fool me somewhat.
@jimweights8908
4 ай бұрын
That is great and I cannot believe I didn’t know this until now!
@PlayNowWorkLater
5 ай бұрын
Nice concise explanation! Really enjoy your enthusiasm
@paulshelley7664
6 ай бұрын
Wow. How exciting indeed! Brilliant
@glenndavis4452
6 ай бұрын
Amazing symmetry. Geometry gets used more than some people think. In my machine shop classes we would have to try and calculate the radius tangent points of a cutting tool to determine perfect tool path and direction. In order to get the desired radius or angle in the metal parts. It got really complex. Fortunately with computer controls, the machines can actually do the micro trigonometry for each individual tool, and we only have to input required dimensions. But understanding the principles helped if something wasn’t running through the metal as needed.
@MrAmitsaurabh
5 ай бұрын
I discovered this solution myself back others are mugging formula and i was thrilled and thought myself a mathematician.
@askmaxim
4 ай бұрын
It's the Pi because the radius is 1. But the "Circle tangent to the three sides" problem also has the 2nd solution: R=6 and area=36*Pi. The circle will be tangent to the "5" side from the outside and to the "3" and "4" sides on their extensions beyond the triangle
@bobsandvegan8222
7 ай бұрын
Andy Math is my favorite kind of math
@Sancho_Merka
7 ай бұрын
It's not just Pi. This is Pi multiplied by a unit of measurement. А circle inside a triangle of 3, 4, 5 centimeters is exactly Pi square centimeters. A circle inside a triangle of 3, 4, 5 meters is exactly Pi square meters and so on. Area is measured in unit squares, Pi is a scalar.
@dielaughing73
6 ай бұрын
He mentions that. But so what? By your logic the lengths '3, 5 and 5' are invalid too, because they're just numbers. It's a pretty well accepted convention in mathematics to take units as assumed
@voidisyinyangvoidisyinyang885
6 ай бұрын
see Philosophy journal 1999 Professors Charles Sayward and Philip Hugly article, "Did the Greeks Discover the Irrational?" The answer is NO!! Why? Because arithmetic distance is not the same as geometric length REQUIRING a "unit of measurement." It's basically Kurt Godel's Incompleteness Theorem as the inverse of transcendental numbers. hahahaha. But also if you understand the foundation of reality is noncommutative then you realize the Pythagorean theorem is a lie from the wrong music theory! the square root of two was first derived from the Major 2nd music interval as 9/8 cubed as the Devil's Interval or tritone.
@dogphlap6749
6 ай бұрын
All new to me, and I managed to follow your explanation as to why it was so.
@bwhog
6 ай бұрын
You could also state it as, "If two tangents to a unit circle are drawn at right angles to each other and a third line connecting these two tangents is drawn such that it is also tangent to the circle, the measurement of those tangents will be 3, 4, and 5 and the included angles will be 30, 60, and 90 degrees."
@davidmyers9480
4 ай бұрын
Don't have a "thumbs-up" on my pallet. My wife, however, says "silly, everyone knows pie are round... cornbread are square."
@Irtsak
4 ай бұрын
Very nice solution . Αlternately ............... Let triangle ABC and BC=a=5 , AB=c=3 and AC=b=4. Semi perimeter of triangle ABC : S=(a+b+c)/2=(5+4+3)/2=6 => S=6 Area of ABC : A=(b*c/)2=(4*3)/2=6 => A=6 There is a formula : A=S*ρ (ρ is radius of the inscribed circle of the triangle ) So 6=6*ρ => ρ=1
@GottabeatemBS
6 ай бұрын
this is class 10 maths and this was the type of sum we get in exam! really helped my comprehension!! please continue these types of videos they are awesome!
@sMeLLwAtER
5 ай бұрын
I once saw a hypersphere of tesseracts .....spinning....it was magical .
@virajelix
5 ай бұрын
Wow! Simple but truly powerful! Thanks a lot!
@flashback-devilsadvocate
5 ай бұрын
im watching this on march 14th...and didn't realize it til just after the video!
@AniBAretz
4 ай бұрын
Very cool. Well presented, too.
@kenttm42
6 ай бұрын
Very elegant proof. Takes me back to middle school (just after the dinosaurs became extinct)
@paulgrimshaw6301
4 ай бұрын
More generally you can work out that a right triangle with sides a, b and h circumscribes a circle with radius ½(a+b-h).
@michaelmello42
4 ай бұрын
Simply ASTOUNDING and right under everyone's nose, as in FOREVER. I can't help but think of the countless teachers and scholars worldwide, including myself, unaware of this beautiful little proof. Can you cite a source for this proof or share a little about how you learned it?
@mikeh1259
6 ай бұрын
That's really awesome! Thanks for showing us that geometric phenomenon. When you say that this works for "all types of 3-4-5 triangle", aside from the one you've shown, what other types are there?
@mikehaddrell8674
6 ай бұрын
Multiples like 6, 8, 14 etc.
@theadamabrams
4 ай бұрын
There aren't any others aside from drawing the triangle sideways or backwards or rotated. To get r = 1 from integer sides you need a = 3, b = 4, c = 5. But actually something extremely similar happens for _any_ right triangle. The radius of the inscribed circle is always r = (a+b-c)/2.
@griffon-vulture
5 ай бұрын
So bright picture with Egyptian triangle and PI cirkle should be on covers of school math textbooks. Thank you!
@flyingofficerayushdwivedi2907
5 ай бұрын
Great, love from India 🇮🇳
@HoSza1
6 ай бұрын
It's not that hard to derive the equality r² = (c-a)(c-b)/2, which holds for any right triangle. (The general case is a bit more complicated but still manageable.) Now a = 3, b = 4 and c = 5, so r² = 2*1/2 = 1, that is r = 1.
@allybally0021
6 ай бұрын
Spooky how precise that is. That is God in the math.
@pickleballer1729
3 ай бұрын
Cool math! Thanks for posting.
@kwaichangcaine8234
4 ай бұрын
I'm a 63 year old Carpenter that was never very good at math. Thanks for the explanation.
@neuraaquaria
5 ай бұрын
Excellent explanation. I started drawing radii of the inscribed circle but took a while to realize the tangent segments theorem. Really cool stuff man.
@frankmccann29
5 ай бұрын
This is the backwards in Time triangle that Dr. Sabine (and Dr. Von Hauser who was actually real and you can't get his work in normal channels) was talking about. Observer position and dealing with getting somewhere before you left due to polar elements. Thanks.
@wolfgangtschirk3542
6 ай бұрын
An amazing fact - and a great explanation! Thank you!
@aliso-pv7ll
3 ай бұрын
Dear Andy, the set of numbers US Congress uses to balance the budget is the set of “Wacko” numbers!
@rowill2968
6 ай бұрын
that is pretty neat!! I'm not sure if or how I will ever use this but I'm glad you pointed it out and its definitely worth a thumbs up :)
@davebasch5995
4 ай бұрын
Fun fact: there are 16 other primitive Pythagorean triples (aside from obvious multiples), with integers under 100... ( 3, 4, 5) is just the first one. My fav is 11,60,61
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