Incircle area in 3-4-5 triangle?

This short demonstrates an intriguing fact about the area of the incircle of the 3-4-5 right triangle. Using two different methods of computing the area of the 3-4-5 rectangle, we find that the inradius has a length of one so that the area of the incircle is Pi!
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To see a related video, showing the general connection between the area of a triangle and the inradius, see • Triangle Area from Inr...
#manim #math #mathvideo #mathshorts #geometry #circle #triangle #animation #theorem #pww #proofwithoutwords #visualproof #proof #incircle #inradius #incircle #semiperimeter #incenter #pi #righttriangle #area
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[@chixenlegjo]

This works for all triangles, too. If you know the area and perimeter, you can find the radius and area of the incircle.

[@MathVisualProofs]

Yes. I have the visual proof on the channel :)

[@john-ic5pz]

however, not every triangle circumscribes a circle of area π ...though an infinite number of them still do and only one of them is a right triangle 📐 🤔 neat.

[@TheBluePhoenix008]

You can just find it with area of perimeter

[@eyewaves...]

Can only work for right angle triangles..

[@coder436]

It also works with any polygon with an incircle

[@KimTiger777]

What I find beautiful with this is the simplicity that it contain soo many things in it. 1 = radius of max sized circle within the triangle 2 = diameter of max sized circle within the triangle 3 = opposite side of the triangle 4 = adjacent side of the triangle 5 = hypotenuse of the triangle 6 = area of the triangle 7 = ??? pi = area of the circle

[@alien_0_0_7]

sum of two sides : 3+4 = 7 3+5 = 8 4+5 = 9

[@alien_0_0_7]

sum of two sides : 3+4 = 7 3+5 = 8 4+5 = 9

[@touchstone1314]

7 = 3 sides + 3 perpendicular bisectors + 1 circle 😅

[@andraspongracz5996]

7 = area of the triangle + 1 8 = area of the triangle + 2 ...

[@PC_Simo]

@@touchstone1314 That’s genius 👍🏻!

[@Invincible2203]

There is also a result for right-triangles that radius of incircle is area of right-angled triangle/semi-perimeter of that right-angled triangle. Same can be observed from that too.

[@moneeshwarborra3915]

Not only for right angled triangle....its valid for every triangle

[@Invincible2203]

​@@moneeshwarborra3915Yes yes, slept through my mind...

[@MathVisualProofs]

Yes. I have that general result on the channel too :) linked in description of this one.

[@WiggyWamWam]

Area divided by half the perimeter? What is the semi perimeter?

[@himesh8001]

​@@WiggyWamWamyes it's half of perimeter

[@dot_lol]

I wish I watched this when I was in geometry

[@reportedstolen3603]

We should start calling this the “unit right triangle” from now on lol

[@bebektoxic2136]

AndyMath told me it was π.

[@vennstudios9885]

if we do it for 5 now it's r(5-(4-r))=r(1+r) or r(r+1) r(5-(3-r))=r(2+r) or r(r+2) then r² combined them into r²+r(r+1)+r(r+2)=6 again simplified r²+r²+r²+3r=6 simplified again 3r²+3r=6 simplified r²+r=2 r²+r-2=0 (r+2)(r-1) WHICH FINALLY GIVES US POSITIVE 1 and followed with the fool the Negative 2

[@matematicaselreyocnkel9229]

you complicated way too much. you can do it with the hyp of 5 knowing that the bisector gives congruent triangles so 5=(4-r)+(3-r) 5=7-2r 2r=2 r=1

[@vennstudios9885]

well we could also just 3-r=5-(4-r) since the half of Kites 3,r is congruent to 5,r and 4,r to 5,r (labels not numbers) then it would be 3-r=5-4+r 3-r=1+r 2=2r 1=r but it is true but I worked to find the trivial answers to R like 6=r²+r(3-r)+r(4-r) 6=r²+3r-r²+4r-r² 6=-r²+7r 0=r²-7r+6 expanding the quadratic it's (r-1)(r-6) which means r=6 is also valid solution

[@the-boy-who-lived]

This is the most basic right triangle we know. And having the most basic unit cricle from it made me surprised. Edit: A 3-4-5 right triangle is special because it is the simplest Pythagorean triple

[@nicreven]

what about half a square that's more basic a right triangle methinks

[@matematicaselreyocnkel9229]

​@@nicreven it does not have all sides with integer lenghts

[@alien_0_0_7]

most basic can be equilateral traingle of side 1

[@the-boy-who-lived]

@@alien_0_0_7 That's not a right triangle

[@alien_0_0_7]

@@the-boy-who-lived sorry I didn't read right

[@braziliangentleman5148]

I used the fact that segments that come from the same point and tangents the same circle have equal length. I ended up with the equation r+5-3+r = 4

[@richardtrager7125]

Same, I used 3-r + 4-r = 5 instead

[@milliosmiles5160]

Awesome explanation, thank you.

[@heardit3115]

In fact, as already mentioned, this works for all triangles, if you know the area A, the side lengths a,b,c, and the perimeter p= a + b + c, you get that A = 1/2 r*a +1/2 r*b + 1/2 r*c, where r is the radius of the circle, which simplified gives 2A = r(a+b+c) => r = 2A/(a+b+c) since the area of circle is πr² =>Ac=4π(A/(a+b+c))² where Ac is the area of the circle

[@user-by5tx6eg5s]

The radius of the inscribed circle of a right triangle is the base + height - hypotenuse divided by 2. 3+4−5=2, 2÷2=1=r.

[@marcelluswallace6240]

I figured it out not via the area but simply by getting r from a system of 3 equations with 3 unknowns. The 3 radii shown disect the triangle into 3 kites and we know that 2 of the adjecent sides of kites always have the same length. Hence you can make a set of 3 equations: x+r=3 y+r=4 x+y=5 solving for x and y in eq 1 and 2 and inserting in eq 3 gives 3-r+4-r=5 7-2r=5 2=2r r=1 q.e.d.

[@oguzhanalhan3521]

You can find it by also saying that the lines that touch the circle are the same length from the corners of the triangle so lets just say that the length of the line that reaches from the corner of 3 and 4 is x and if you write the lenghts as 4-x, 4-x, 1+x and so on you will find that 1+2x=3 and therefore x=1. İf we draw a line from the centre of the circle towards the lines of the triangle wher it touches the circle those lines are have a 90° ont he sides. So there is a square there. Therefore x=r=1

[@mareau2193]

I worked it out by realizing that you had to be able to fit two of those circles inside a rectangle of 3x4, since one circle fits inside half of such a rectangle. Which means the diameter of the circle was half the base, or 2. Radius is half of a diameter, giving a radius of 1, giving an area of pi.

[@jesecamilo7694]

I had a similar insight! I love the many ways to approach such problems. Geometry (math in general) is beautiful that way :)

[@thegamingnerd274]

Ima be honest, I said pi for the loss and wasn't expecting to be right

[@Tekyng_of_Baregan]

Same:)

[@BuggaUgga]

That's some really nice mathematics here

[@birukshegute5035]

So beautiful. You deserve a subscription.

[@evan.z974]

*3* .1 *4* 1 *5*

[@cyberrgg6452]

Lol

[@Muizur]

6 if rounded properly

[@kensurrency2564]

good insight

[@supercrafter7586]

​@@Muizurhe didn't show an approximation, he showed a part of pi, there's no approximation sign here

[@gdhammr8113]

I was literally thinking “please be pi” and was pleasantly surprised

[@baoboumusic]

pi is more than 3 so that could never work.

[@abehankens7456]

thats super neat, especially from such a random triangle like the 3 4 5 pythagorean triple!

[@GusBatista03]

π hides in some truly mind boggling places😮😁

[@bananaforscale1283]

People when a circle constant appears in circles: 😲😲😲😲😲

[@scoutgaming737]

Bro it's a fucking circle of course there's pi

[@GusBatista03]

@@bananaforscale1283 Alright, you got me there🙄

[@GusBatista03]

@@scoutgaming737 What's surprising is that the unit circle fits perfectly in a 3/4/5 right triangle

[@octs609]

Shows how much smarter you think you are then actuality, he clearly is surprised because the area is perfectly pi.​@scoutgaming737

[@kaninma7237]

I love it!

[@MercyTN581]

That is just beautiful

[@quokka_yt]

Another way we can go about this First, we find the surface area of the triangle: S∆ = (3 ⋅ 4) / 2 = = 6 Now we find the semiperimeter: p = (3 + 4 + 5) / 2 = = 6 Now we use this formula: S = pr 6 = 6r ⟹ ⟹ r = 1 Now we use the formula for the area of a circle: So = πr² = = π ⋅ 1² = = π

[@ganchos2849]

It really was interesting! Thank you!

[@Ki1aGam3z]

Not me watching this during my free time

[@bishiyarudesu]

Another way to find in radius would be to equate length of tangents: 4-r=5-(3-r) 4-r=2+r r=1.

[@johnbutler4631]

Thats really cool!

[@LauraEdgar-xv1bs]

Great vid, yo!

[@vuk5001]

r = a+b-c/2 r = 7-5/2 r = 1 P = r²π P = 1π = π Thats how i got it, using the a+b-c/2 is way quicker

[@mathmachine4266]

(r,r) is the center. The distance from there to the hypotenuse is the dot product between the normal vector and a line between the center and literally any point on the line. • = 12/5-3r/5-4r/5=12/5-7r/5. 12/5-7r/5=r, 12/5=12r/5, r=1. The radius is 1, the area is π.

[@izzabelladogalini]

WTAF its so obvious when described like this but it's never occurred to me before ... Love these little vudeos, keep it up please

[@-ShootTheGlass-]

I’m not entirely convinced that’s even remotely useful to me personally, yet I’m mesmerised by these clips. 😂

[@jesse_cole]

In the words of ICP... "Miracles."

[@naved5625]

Another simple solution is :- 1. The square that is made has length = r 2. So the other sides of the square will also be equal to r 3. Hence we can write that The length of tangents = 3-r and 4-r 4.by tangent theorem we know that we can write the hypotenuse as 4-r+3-r = 5 so 2r= 7-5 hence r= 1

[@tepochcalli]

That's how i solved it in my head. Much simpler

[@neologicalgamer3437]

Well, I don't think the pi is too impressive, seeing as we're literally multiplying by pi, but I certainly think that's wild that we get the radius to equal such a perfect 1 in such a popular triangle

[@Im_Ish]

Pi: "What up, my fellow right triangles?"

[@aminramazanifar9743]

amazing!

[@twickersruss]

The reason why pi is involved is because you drew a circle effectively introducing circular considerations.

[@user-gc1hk1vv6k]

thank you thank you...

[@PC_Simo]

I’m approximately at the 1/3-mark; and yes, I *_DO,_* in fact, know the area: π (A = πr² ∧ r = 1 -> A = π*1² = π) 🟢. 🙂

[@hayden4673]

I’m not gonna lie, i just eyeballed the radius to get the answer

[@dylantd9189]

I just was like "that radius looks like a 4th of the 4" and was like 1πR² so π 😂

[@mariec7958]

Another prove that math is beautiful ❤

[@3eH09obp2]

Genius!!!

[@khiemoan2894]

For any triangle: S = pr, with p represents half circumference of the triangle and r represents radius of inscribed circle

[@Guy_Sandler]

Of course I see this AFTER my test on this unit

[@ranua9327]

Awesome! and comments as well!

[@megajava505]

Formula for this triangle r= (a+b-c)/2

[@mach1553]

3+4-5=2 ... 2÷2=1

[@vgrants1717]

Me: “It looks like it’s about 2/3 the way up the 3 side and half the 4 side. Diameter of 2 = radius of 1 = area of pi.”

[@mach1553]

The magic triangle 📐

[@cccexestarted]

I'm almost dying from amazement 🤯🤯🤯

[@TheAllanmc64]

Looks like a Starfleet Communicator pin

[@HighMojo]

I'm not following the computations, the explanatory animation is just so amazingly choreographed.

[@PraneethP-qf1yt]

There is another way. The three vertices of the triangle each give two tangents touching the circle. We know that tangents from same point have equal length. And we see a square formed by two radius and twk tangent . With this we can find r length.

[@afraid2letgo]

A cooler way to find the radius of an incircle in a right triangle is: (cathetus + cathetus - hypotenuse) / 2 = (3 + 4 - 5) / 2 = 1

[@tomadshead6133]

Another simple solution is find the centre of mass with ⅓[4,0] + ⅓[0,3], so cofm is (4/3,1), meaning the radius down to the bottom is also 1 so area is πx1² = π

[@Doodle1283]

I was always amazed about how the incircle's radius of the simplest pythagorean triangle turns out to be 1. Never thought about pi though😂

[@eyewaves...]

Interesting indeed!!!

[@jackdog06]

My brain: “Idk it looks like the radius is about 1 and an area of pi is probably interesting enough for a KZbin short so I’m gonna say it’s that”

[@usamathtutoring]

The unit circle 😮

[@lepulsar21]

Wait until Poncelet hears about this one

[@thomasrebotier1741]

Or simpler, the 6 triangles defined by the bissectors are 3 pairs of symmetric triangles and their sides a b c add up a+b=3 a+c=4 b+c=5 obviously a=1, and r=a.

[@SyllabusTheWise0047]

Math makes me weak in the knees

[@Secret_Moon]

Or you can just use (3 + 4 − 5) ÷ 2 to get r = 1.

[@Clyntax]

Better visual proof would be to rotate the 3 and 4 sides onto the 5 side. You will see that the overlapping part is exactly r which has to be 1 as 3+4 is 5+2. pi* r^2 gives pi as area.

[@user-sy5ok7hw2v]

Pi is like the friend who comes to every party without invitation 😂

[@xys007]

The pie recipes are getting weirder and weirder.

[@paulborst4724]

Why? Use geometry instead. p = 3 - r, q = 4 - r, 5 = p + q, thus r = 1 & A = Pi.

[@mab9316]

A new definition of Pi

[@krishgarg2806]

luckily I just remember the formula r = Δ/s

[@sp1xa]

also it can be proven that R = P - a = (3+4+5)/2 - 5 = 6 - 5 = 1 S = πr² = π

[@soubhagya8808]

Inradius= ½×(Perpendicular + base - hypotenuse)

[@soubhagya8808]

» r = ½(3+4-5) = ½×2 = 1 » πr² = π

[@markphc99]

yes , it is interesting

[@TheDanioneal]

I just guessed pi because i thought it would be a fun answer that made sense, glad to know my sense was correct lol

[@JaesulHwang-oc3zj]

r=1 because (3-r)+(4-r)=5. so circle area= 3.14

[@Ms.Cookie]

Pi is a really cool mathematical constant, I think people think about it too much as just being a thing with circles; of course all things related to pee are also related to circles because Pi is related to circles but all math is related to other Math so

[@purpleballooon]

my thinking: “well r looks like 1/4th the length of leg 4 so since 4/4 is 1 then the radius is 1”

[@captainwasabi]

I did this knowing that the sub right triangles would be congruent. This means that 3r = 4r => r = 1

[@ValkyRiver]

Mathologer also demonstrated how this is related to Heron’s Formula as well as 1+2+3 = 1x2x3.

[@Georgeclassified]

Just use Viviani's theorem... we have a right triangle so the height is 3 and since the center of the circle is an point within the triangle and each radius r is prependicular to the side with which is shares tangency, the three r must add up to 3 so we have r+r+r=3 so 3r=3 so r is 1.The area of the circle will therefore be π1² so 1π or π

[@MathVisualProofs]

Viviani’s theorem is for equilateral triangles. I think it’s coincidence that the numbers work here.

[@jimi02468]

Draw a circle around an isosceles right triangle so that the circle goes through every vertex. The ratio between the area of the circle and the area of the triangle is pi.

[@8b25arjunarvind6]

Nice

[@Scubadooper]

Pi from Pythagoras

[@sandeepdrebel233]

Inradius= (p+b-h)/2

[@Snakles08]

I’m tired of pi showing up everywhere, who invited this guy?

[@nilsalmgren4492]

Nice use of algebra.

[@Mediterranean81]

3-r+4-r=5=>7-2r=5=>-2r=-2=>r=1 by the two tangent equality

[@burrdid]

I guessed it!

[@voidey_]

It's π.r² for sure

[@duyminh2803]

a+b=5 r=3-a=4-b => 2r=7-a-b=7-5=2 =>r=1

[@visheshsomani9731]

You can prove that radius of an incircle of a right triangle is the([ sum of bases] - hypotenuse) /2

[@htvz]

Existe el Teorema de Poncelet para calcular el radio

[@mrwess1927]

Wow

[@stella6081]

this broke my brain

[@meh4990]

Random recommendation, but it might be useful!

[@timokimo8206]

🤯

[@paulclissold1525]

wow

[@ivanmehboob]

If the sides are 27,64,125 would it be π^3 or would it still be π

[@Mag1p0rp0ra]

My friend, that is impossible. The longest side is more than the sum of the other 2, so it cannot be a triangle. If the two catheti are 27 and 64, hypotenuse would be 69.46. :)

[@ivanmehboob]

@@Mag1p0rp0ra ok so its just multiples and not exponentials