I've always used the law of cosines to prove it, but this is pretty slick! Thx bprp

"HE RUNS" formula KZfaq's captions in a nutshell

I love how he pronounces “cancelled” as “canceldid” so much

Thats a nice proof without any trigo involved making it clean and simple😄

When you kept playing with the factorization rules at around 6:00, I already figured out how to prove Heron's formula. I tried to verify the formula years ago using the sine rule (A = ½bc sin t), but the equation got very complicated until I didn't know how to simplify it. This video shows the importance of mastering algebra, especially when it comes to solving simple problems like this.

10:38 Never heard of that, but COOL!

Thats great video i always thought this theorem was long and needed so much effort so i never been curious about it and rarely used it but you changed my mind keep up the good work

I got asked to do this as an interview question It took some, to say the least...

Your tone is really great - that is half the battle of being a good teacher. Great video I enjoyed it.

I’ve been wanting to see a proof of this formula for a while now. Thanks for showing this great proof!

The formula for area of quadrilateral was shocking. Wow! Good information. You are doing good.

Thank you! I’ve always wondered about the proof!

Perfectly explained. Loved the video. Thank you so much😊

I proved Heron's formula a few years ago with SOHCAHTOA. This proof is much nicer and more concise. Great video, BlackPenRedPen!

u r just great, thanks for making our studies easier, soon my exams, and so blessed to have found ur channel)))

I'm so happy I found this, stay safe

I'm a backbencher sir,but your every explanation is just so easy to understand ♥️

Cuuute! it's something even young students can do to really stretch their algebra skills hehe it's easy but with some algebra tricks 😊 nice

The formula is introduced in Heron's book Περί Διόπτρας, where he proves it by using the inscribed circle, an elegant geometrical proof

I never knew about this formula, and the proof is really easy but I found this video extremely entertaniing

I've squared the second quantity under the root and struggled with the algebra but finally I looked at what I had which is a fourth degree polynomial in terms of "a" and solved for "a" squared and took the square root and rearranged the solutions to get the product of the final 4 quantities Really amazing problem that I can actually solve.

I enjoyed very much. Thanks for making such nice videos!

Wow, I'd never heard of Bretschneider's formula at 10:38, that's weird! How do you prove it? It reduces to Heron when d=0.

Thank you I've been dreaming about learning proof of this formula some 5 years now

Presh Talwalkar's fans will be complaining of you not using Gougu's theorem :-) Anyway, you are the king of KZfaq math-teachers!!

I love youuu, so helpful, u just expplained it so simply and clearly

Blackcursorwhitecursor

Great video! Please do more proofs!

This video is absolutely perefect form my math project!!!!! TYSM!!

I've come up with a formula for the area of triangles using hard algebric geometry. It takes the sides squared as inputs, so it works best on a carthesian plane. A,B,C are sides squared A=1/4 * sqrt(- A2 - B2 - C2 + 2(AB+BC+CA)) it uses pretty big numbers so it's better to use a calculator or use it in a program... But I'm sure it can be transformed into heron's and viceversa.

I was just curious as to how this was derived and this derivation is neat!

I love this proof. Pls make more videos like this

Very Easy to understand...Thank you

Nice work. In the generalized Brahmagupta’s formula angle aplha correctly is half of sum of opposite angles.

Beautiful. A really excellent explanation.

I actually discovered this formula in religion class by accident when I was playing around with 1/2ab * sin(C) and cosine rule (to find the angle used in the area formula and then use inverse trig identity). Thankyou for sharing this.

We can use cosine formula a2= b2+c2-2abcos(c).Work out since and use area formula A= 0.5absinc

Old School Style blackpenredpen!

Awesome explanation!

Good bro, your videos are amazing. Please try in upcoming videos to solve Derivative of x!

thanks for the amazing content

Hi, on Wikipedia it says that Heron originally proved this using cyclic quadrilaterals; please could you make a video on that? Thanks so much.

By me best herons formula prof is to prof volume of equilateral triangle , and then any other treangle as resized equilateral in two directions, so this can be used as prof for n-dimensional triangles volume

Please do a video explaining the Bretschneider's formula at 10:38

احسنتم وبارك الله فيكم وعليكم والله يحفظكم يحفظكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين .

I love your videos! Can you tell my what programs do you use to record the screen and what app/program do you use to write? Do you use mouse for writing?

In Brahmagupta's formula I think θ is the sum of two opposite angles divided by two. I really like this video.

What 3B1B is a patron? Damn!

The same way Heron's formula works for triangles and Brahmagupta's works for quadrilaterals, I wonder if there's a general pattern for any polygon with n sides. I assume that the proof for the quadrilateral formula comes from cutting the quadrilateral into 2 triangles and applying Heron's twice, so theoretically it's possible to derive a formula for a pentagon and so on.

That was beautiful!

Thank you for this great ful video

It's really good sir.i want more mathematical proof sir

I love the phrase "invite into the square root house", I never thought of thinking of it that way.

My man please prove Stewart's theorem.

When I was in 9th there is where I learned heron formula & as note I found brahmagupta's formula & I'm amazed that just putting d=0 you can get heron's equation.. Man Indian Mathematician were too good at that time I always love to learn more & more about them

3b1b is one of your patrons? That's awesome!

Damn I always forget about Heron's formula and it's so useful! I totally could have used this on my Calc 2 homework a few weeks ago

Such a gorgeous proof ✔

There's a simpler version. Through law of cosines, we have cos(A)=(a^2+b^2-c^2)/2ab. Then, we have sin(A)=sqrt(1-cos^2(A))=(1-cos(A))(1+cos(A)) and you can easily finish the proof using 1/2 bc*sin(A). It's the same algebra as above except you skip a lot of steps.

I always wanted to know this.

Hi, nice video buddy! Can you answer me this question? How many planes are defined by one line and 3 collinear points that do not lie on that line

The same way i also derived this formula .....It's suprising to me that I can think like the blackpenredpen....

Great, this video proved 3 formulas at the same time, one formula attributed to a Chinese mathematician from the 13th century, then a formula found by Kahan anb finally the Heron's formula.

splendor, bro carry on

My most favourite proof.

Hey! Great to see your proof of Heron's formula. The way I know is based on formula A = bc*sin(a) where a-(alpha) is angle between sides b and c in a given triangle. Would love to see more geometry on your channel.

or Area=1/4 sqr((P(P-2a)(P-2b)(P-2c)) P is The perimetr of ABC

thank you so much!

YEEESSSS. I LOVE IT. Now how the hell did Heron ever figure that out?

Finally a proof I understand: :’)

heya! can you make more videos like that logarithmic one? im home... in boredom and I love solving problems like that ^^ thx

@bluepenredpen I have a doubt well, are all numbers equidistant from infinity?

Now at 10:40 how about generalize for any n gon or if that's too hard the next level - area of a pentagon in terms of semiperimeter.

The first part of the proof is so simple and straightforward yet I have never been able to do it on my own (maybe I did the first part of the proof, but I know for sure that I was never able to prove this formula which bugged me since I always feel uneasy using formulas that I can neither prove rigorously or have some good intuitive understanding why they should be true without knowing the rigorous proof. Just implementing/using a formula that I have read in a textbook always felt like cheating)

Thanks!

Then, is it so that the circumradius of a triangle is related to its area using heron's formula??

Sir you are very talented

In any Δ ABC, the Cosine Rule gives cos(C) = (a²+b²-c²)/(2ab). So, sin(C)= √[-cos²(C)] =√[(2ab)²-(a²+b²-c²)²]/(2ab). ∴ area(ABC) =(ab/2).sin(C) =√[(2ab/4)²-{(a²+b²-c²)/4}²] which can be facto -rized to give Heron's formula. But who need's Heron's formula! For the 5,6,7 triangle; the area = √[{2(5)(6)/4}²-{(5²+6²-7²)/4}²] = √[(60/4)² - (12/4)²] = √[15² - 3²] = √(225 - 9) = √216 = 6√6. .

Wow amazing formula 😍

Nice prove

thats so cool!!

I just Love it.

Whats brahmagupta's formula Yeah ik it gives the quad area but pls elaborate it

I'm going to use this for right triangles from now on and nobody can stop me

pls do the trigonometrical proof also using sine and cosine formulae

It's quite complex but alongside its helpful too! :⁠-⁠)

thank you !!!!

Thank you

I was at grade 8 when my teacher asked to prove heron's formula for a triangle's area. Idk but I somehow managed to prove it, now, three years later, I can't even do simple banking problems (SI and CI

Embezzlement can be very well demonstrated using geometry

1/2*a*b*sin(C) is much simpler imo. (C is the angle between the sides a and b) you can easily derive it geometrically if you draw h on the side a: sin(C)= opposite/hypothenuse = h/b so h=b*sin(C)

At 3:18, you can do a bonus proof of the law of cosines easily by seeing from the picture that b₁ = a cosγ and rearranging. I don't know how I missed this one back in April!

10:39 this isn't the Brahmagupta Formula, this is the Brettsneider formula, in the Brahmagupta, there isn't the (-abcd*cos theta) however it work just on cyclic quadrilateral. Except this, the proof is good.

ThanksMaster

It can be derived from a particular case of the generalized half angle formula. Se here: kzfaq.info/get/bejne/jcibhKufs9rMl2w.html

Less operational is with the half angle formula that involves the semi-perimeter

I am in 10th grade and this is the first video of BPRP that I understood well

Now do the formula for the area of a pentagon.

Nice upload

At 10:38 shouldn't theta be the average of the two opposite angles, rather than the sum? That's what wolfram says, anyway. It's also the only way to get the right area for a square