Finding the Area of a Non-Right Triangle

[@Dt.Dicaro]

Thanks Heron

[@arta.xshaca]

He didn’t invent it. Just like Pythagoras

[@Dt.Dicaro]

@@arta.xshaca it's still cool

[@bashersinc.1715]

how to prove herons formula?.

[@okicek3016]

​@@bashersinc.1715there's a bprp video on it

[@panagiotisathanasiou7918]

Greeks invented everything

[@farhanfouadacca]

Never heard of Heron’s formula before. Thanks

[@mariosantos203]

The important thing is the logic behind the formula/procedure. I just don’t understand why math teachers kept on demonstrating the solution to a problem without explaining the logic behind. I suspect they don’t understand too.

[@mrhtutoring]

Give me 60 minutes. I'll explain it so that even an elementary school student would understand why. It's a tough squeeze just to mention the formula for KZbin shorts.

[@suthan2003]

He definitely knows it. But sometimes it is sufficient to just know the formula. Example I don't need to know how Engine of the car works but as long as I can drive I am fine

[@mariosantos203]

@@suthan2003 if the purpose of maths in school is to train the mind it certainly a must to understand the logic behind the formula.

[@anonymoushere7786]

​@@mariosantos203derivation for every formula is not needed imo

[@TikeMyson69]

@mariosantos203 You might've not noticed but this is KZbin, not school.

[@anestismoutafidis4575]

Thank you for showing the Heron's method. At first , I calculated the area of the triangle by calculating the height of it (a•sinβ) and then by the modificated formula for calculating the area of the dragon figure ( A= 1/2 •e•f)÷2 9,8m^2. It took a relativ long time, of course. I am grateful to know now about the Heron's formula.

[@Nino-eo8ey]

where'd you get the angle from

[@yessssssh]

​@@Nino-eo8eyBro you don't get sarcasm do you?

[@Nino-eo8ey]

@@yessssssh You can't blame me, thought he actually measured the angle by himself when he said "took a long time, of course".

[@anthonylawrence3265]

That is absolutely brilliant, I don’t know why it never occurred to me to perform the operation in this manner.

[@alexandrevaliquette3883]

Meanwhile, some flat earthers struggle with basic math to figure out the shape of earth!

[@Mustafa_Shahzad]

Blud what This is pure maths, that would be a specific example of a certain specific field of applied maths. Like I think your comment was a bit of a stretch. Flat earthers can definitely be good at mathematics aswell.

[@alexandrevaliquette3883]

@@Mustafa_Shahzad Most (if not all) of them cannot calculate basic stuff. For example, they cannot calculate the distance between two point on a sphere (regardless if it's earth or a sphere draw on a board).

[@seventhunder777]

Where was Heron when I was in school?

[@NinjaBear1993]

It is trigonometry/ pre calculus. I learned it in trigonometry in college. I was a sophomore in high school when I took it.

[@GlavnyiGitarist]

Wow. We were learned Heron in 8th or 9th grade

[@maxpie16]

@@GlavnyiGitaristsame

[@taimurahmad]

I learned it in grade 7

[@nstarling88]

As a math teacher I forgot this😔.

[@LiquidRR]

Me too!

[@johnplong3644]

You just made me feel better I totally forgot this I tutored kids in basic math up to Algebra 2

[@dilaudid1]

Wikipedia explains it's derivation. You can figure it out using the Pythagorean theorem making this triangle into two right triangles, with the shared side = x, and the 7 meter sides = y and 7-y, etc etc. Its an elegant formula from thousands of years ago.

[@ieo8446]

This seems awfully useful for something that's not taught in schools

[@Verziroo]

I guess because working out the area of a triangle using 1/2•bh or 1/2•ab•sinC (when height is unknown) is more applicable to other branches of maths whereas this doesn’t really carry over to any other branch. Also more time-consuming than any of the two mentioned. Still nice to know though

[@mrydobon]

It's also not applicable to all triangles. The sum of any two side lengths must be greater than the third side length

[@cmk4489]

Literally used this two days ago in an exercise in numerical mathematics! The task was to program the triangulation of a circle to compute its Area

[@xdragon2k]

I just noticed that for a three-sided polygon to be called a triangle, the sum of it shortest sides need to be more than the longest side. It can never be smaller than. And if they are equal, it's just a straight line.

[@oc1anking956]

no offense but thats kinda basic- a person who couldn't talk properly until 7

[@xdragon2k]

@@oc1anking956 I know it's basic. It's just something you don't think about. That is until they show you this formula. S = the sum of all sides divided by 2. Then you multiply S with S minus each sides. On the straight line case where a = b + c, s - a will equal zero. A straight line has no area.

[@oc1anking956]

@@xdragon2k ahhhh I agree on u with this one

[@mikaeloverfjord9047]

Deriving the formula yourself you will see how elegant this formula is. (You will have to know the Area Formula A=ab*sin(C)/2 and the trigonometric identity sin(x)^2 + cos(x)^2 = 1 and the Law of Cosines cos(C)=(c^2-a^2-b^2)/(-2ab) )

[@DamplyDoo]

I actually was the one who told this to Heron. Bro took credit

[@wolemai]

That is one weird formula! Later you may want to show how they derived that formula - not now, I don't have the time!

[@ernestobie-wm1ei]

Thank you prof. You are the best 👌

[@jpc70000]

Thank so much

[@loridave1962]

Wow. I need to think this thru to see if it should be intuitive. Way cool.

[@Omnipotent_Science]

W heron let us do simple arithmetic 😎

[@Power10180]

I think Heron's formula is the best expression in geometry

[@420sakura1]

Root of 96. 100 is Close to 96. Sqrt of 100 is 10. 10 - (100-96)/(3*10) 10-0.2=9.8 Sqrt of (96) it's 9.7979

[@junesantos543]

I'm confused too!.. square root of 96 arrived at 4 sqrt of 6?

[@junesantos543]

Now I knew, simplifying radicals.. sqrt of 16 X sqrt of 6 = 4 x sqrt of 6

[@sachinbarvekar2563]

4/20 (100-96) (10*2)

[@fubaralakbar6800]

What if you inscribe this triangle into a rectangle, compute the area of the rectangle, compute the area of the two right triangles outside the non-right triangle, and then subtract? Ah, but I suppose you can't do that without knowing the height of the triangle, which is impossible to compute with the information given.

[@carultch]

It's not impossible to compute with the given information, it's just requires a detour from algebraic operations. The solution ends up being an algebraic function after all, since it is a composition of sine and inverse cosine. Let A be the angle opposite the 5-meter side From the law of cosines: 5^2 = 4^2 + 7^2 - 2*4*7*cos(A) Solve for cos(A): cos(A) = (25-16-49)/(-56) = 5/7 The height h will equal 4*sin(A). Use the identity, sin(arccos(A)) = sqrt(1 - A^2) to compute it. h = 4*sqrt(1 - (5/7)^2) Thus, the area of the triangle is 7*h/2, which is: Area = 14*sqrt(1 - (5/7)^2) = 4*sqrt(6) Area = 9.8 m^2

[@Mustafa-oy7ly]

You could also do cosine rule to find one angle then you could do 1/2 ab sin (c)

[@hassankazimaziz6402]

Hi sir , what kind of projector that you are using? Thanks for these incredible videos

[@mohammadchaudhry4728]

Itis the shortest way of solution and really very smart way of solution. Of course my mathematical knowledge is very very old since 1969,but if we calculate the attitude of this triangle we can calculate the area of the triangle. Altitude multiplied by base of the triangle ( here it is 7 m) divided by 2 . But the gentleman man has suggested is the shortest method of solution.

[@evinos9310]

we in india learn this formula in 9th grade ie when we are 14

[@Aerospace202X]

I still remember my old school math classroom❤❤

[@bradellis-cole4845]

Nice

[@andersongomez636]

Knowing this is cool …prove this is the way!

[@lawrencelawsen6824]

This guy is god I just subscribed to him

[@stroescualexandru8514]

It is known as the formula of Heron

[@joseacosta1354]

And he retained the units.

[@Snownam227]

how can a formula so simple that 90% get it wrong

[@JeevanPradhanG1]

My question why divide always with 2 wanted to know the reason for it

[@user-iv3in2ou3p]

It feels almost like cheating... Ahahaha. That is education is.

[@pixonicpredator]

Can someone please help me understand the logic behind this formula?

[@karcha5]

Thanks Brahmagupta. He was the one who discovered "Heron's Formula"

[@kaeez]

Don't care, I'll still call it Heron's Formula.

[@karcha5]

@@kaeez Read "The Imperishable Seed". Well, if not you at least others who "care" will read.

[@kaeez]

@@karcha5 Dude, there is no such thing as an "inventor" in Mathematics. There is nothing to be "invented" here. Mathematics is us humans trying to make sense of things already present in the universe. Patterns, repeated occurrences etc. Same with all other sciences. Especially in the case of Mathematics, fighting over who invented something is absurd cause if not them, someone else will one day. It's omni-present, you just have to figure it out. Brahmagupta might have figured this out earlier but it was Heron's publications that were popularized and made mainstream. You can't say Heron copied it off Brahmagupta cause it's not like these patterns were only revealed to Brahmagupta by some extraterrestrial species or God if you allow it, if not him, you could've also discovered it. So really this argument is just irrelevant.

[@karcha5]

In fact Brahmagupta gave a generalization to cyclic quadrilateral (if one of the sides is substituted as zero it's a triangle and his formula then reduces to Heron's Formula). Thus the reading I recommended is perfectly relevant. If you want go read and find that formula, or discover it yourself.

[@kk12181]

Isosceles triangle!!

[@arqmark]

Don't give us the formula, give us the deduction to get the formula.

[@composerlmythomorphic2635]

4^2-x^2=5^2-(7-x)^2 0=25-16+x^2-49+14x-x^2 14x-40=0 x=20/7 Area=(20/7)(7)/2=10

[@mogalephistus3230]

Kind of Simpsons rule💯💯

[@frankiesosab]

u can also find cosx and then area = 1/2ab*sinx

[@johnplong3644]

God I totally forgot this yes I definitely need to take trig again it has be 45 years

[@artemirrlazaris7406]

How did Heron find out this method? What relation of ratio did she realise, Its divded by 2... I haven't checked the answer, but 360 degrees, is it assuming its a square when adding them together. Typically geometry gives you a formula that hten you cna plug in. One triangle thing is 4 times 5 = 20.. so we know the area, will be less than 10... its the hypotenuse that is acting as a modifier causing it to be a non right triangle... Does this work for obtuse only? or narrow isoceles... hmm.. curious on the proof for it.

[@notayush2064]

Heron is the truehero

[@dan_the_man4401]

Would like a proof

[@Mecal21]

Why can't you just do 1/2bh?

[@kiburamaha7715]

Thank you sir! 🙏 One question? Is there a derivation or a proof for the heron's formula?

[@bashersinc.1715]

yes it is a long process...start by the right triangle

[@mohammadhuzaifkhan7828]

How to derive it

[@godstiniestlover801]

Hello sir, is that the method derived from the x-y axis ?

[@ThatGithubGuy]

I thought that was only for triangles whose points lie on a circle?

[@rivenoak]

all triangles work on a certain circle. perhaps not so ancient wisdom as Heron of Alexandria's formula here but old knowledge still for construction of such a circle a look at wikipedia will suffice :D en.wikipedia.org/wiki/Circumcircle

[@KnowYourself06]

I am having the same doubt. This was taught with drawing a circle touching all points on the triangle.

[@ThatGithubGuy]

@@rivenoak Uhh, not all triangles fall in a circle. For example, no isosceles triangle would ever fall on a circle, as it would have to be an ellipse to fall on it. the 7m divided by 2 must be equal to the height, thus the height, if this was true, would be 3.5? I'm not sure if that triangle would fall on a circle.

[@whoff59]

Thinking a triangle and whether all 3 points are on a circle: Each of these 3 points must be equidistant to some point in the centre of the circle. This distance is called the radius. You can draw a line between 2 points and cut it in the midst by another line rectangular to it. Every 3rd point which is also equidistant to those 2 points must be found on that line. You can easily find 3rd points which are not on that line. So you are right: there are triangles which do not fulfil the requirement.

[@rivenoak]

@@ThatGithubGuy naaah, consult the link above; it will work out just fine :D

[@Kamabushi999]

Not done yet The answer has to be in m^2 To be in anyway useful 9.8 m^2 4xSR of 96 is 9.8 m^2

[@yasaissogood]

I don’t understand last part where he simplify it How did he get 4 square root of 6m squared I never seen this in grade seven

[@nabelalifxiiak4565]

√96 = √16x6 = 4√6

[@Hardiarm]

Why would you simplify SQRT 96 as 4*SQRT6?

[@mrhtutoring]

Most teachers want it like that.

[@adrianpaligar2026]

What happened to you FB page?

[@leewmlee]

This is new to me! THKS!

[@mineshukri3444]

Never understood this never wll

[@JezzBowden]

How did he get 4root6 from root 96? I know it’s the same answer. I can use a calculator lol. But how do you make that conversion?

[@mrhtutoring]

√96 = √16×6 = √16 x √6 = 4√6

[@JezzBowden]

@@mrhtutoring Ah, I see. So any simplification will do I guess. Whatever makes it easier to calculate. Because 96 can be broken down in many ways!

[@aroshneupane2462]

Many people are saying they didn't learn this at school but i actually learned it at class 9th.is it really not taught.

[@gamtax]

Any reason why Pythagoras' Theorem overrated compared to this?

[@elmersbalm5219]

Pythagoras can be used to derive a proof. Now I half remember that one, after checking Wikipedia.

[@jakegibson718]

Hi, how would you solve the equation a(x-b)(x-8) Vertex point is 2,-54 and graphed on XY plane find a+b I couldn’t figure out this for the life of me

[@carultch]

Given: a*(x - b)*(x - 8) Vertex: (2, -54) For parabolas in general, the vertex's x-position is average of the two x-intercepts. This isn't true for polynomials in general, but for parabolas is is. This means that the average of b & 8 is +2. We can determine the value of b: (b + 8)/2 = 2 b + 8 = 4 b = -4 Thus, the equation of the parabola is: y = a*(x + 4)*(x - 8) Plug in our known point of x=2 and y = -54, and solve for a: -54 = a*(2 + 4)*(2 - 8) -54 = a*6*(-6) a = -54/(-36) a = 3/2 Thus, the equation of the parabola is: y = 3/2*(x + 4)*(x - 8) And a + b = 3/2 - 4 = -5/2.

[@jakegibson718]

@@carultch Thank you so much this was really bugging me, I greatly appreciate it!

[@majdirabieljilani9675]

why does it work

[@elsenored562]

½ the perimeter = 16 ÷ 2 = 8 A = √(8 × 4 × 3 × 1) = 4√6 m²

[@junesantos543]

Square root of 96 = 9.7 | how did it arrived to 4 square root of 6? Just confused..

[@junesantos543]

Now I knew, simplifying radicals.. sqrt of 96 = sqrt of 16 X sqrt of 6 = 4 sqrt of 6

[@hasantao]

This is not a good way to teach math. You need to learn why not how.

[@user-xb1ht4py2v]

is this derived from Pythagorin ??

[@bashersinc.1715]

yes

[@fascistpedant758]

How do yet you get from root 96 to 4 root 6?

[@mrhtutoring]

√96 √(16×6) √16× √6 4 √6

[@fascistpedant758]

@@mrhtutoring Thank you.

[@AhmedShaban-ju8rf]

Where is the proof

[@Just_Shadow-_-]

sorry but couldn’t i have just pretended that 4m was the base of the triangle and so 5m was the height? this was i could just have done (4m•5m)/2

[@rivenoak]

only works with rectangular triangles. in those the two shorter sides work as base and height for each other and for area size it is also obvious because such a triangle is just a rectangle split diagonally. area of the rectangle would be product of two different sides and the triangle is half the size.

[@whoff59]

The definition of "height" in a triangle includes a right angle at the base.

[@majdirabieljilani9675]

5 is not the height

[@AhmedShaban-ju8rf]

Please don't present mathematical theorems without proofs respect mathematics

[@piccolo64]

i don't know if exist but i found by myself this formula A= √[(2ac)^2 - (a^2 + c^2 - b^2)^2] / 4 , not elegant but is working 😮 i supposed two circles with center (0,0) radius a and (c,0) radius b, the intercept is (x,y) where y will be value of height and Area will be c*y/2, start is system { x^2 + y^2 = a^2; (x-c)^2 + y^2 = b^2} and after some calculus i found this formula. if i choose a=4, b=5 anc c=7 then i use formula and: A= √((2*4*7)^2 - (4^2 + 7^2 - 5^2)^2)/4 = √(3136 - 1600)/4 = √1536 / 4 = √(256*6)/4 = 16√6 /4 = 4√6 cm^2 sorry i'm french, my english is very bad ^^' edit: after factoring and some manipulations, i get the heron formula. Maths are beautiful.

[@composerlmythomorphic2635]

4^2-x^2=5^2-(7-x)^2 0=25-16+x^2-49+14x-x^2 14x-40=0 x=20/7 Area=(20/7)(7)/2=10