Triangle area formula

This short video shows how to prove that the area of the triangle is given by the product of the inradius and the semiperimeter. Here is a longer, alternate version: • Triangle Area from Inr...
Thanks to Colin Wright for suggesting this particular visualization.
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Music in this video:
Prelude No. 9 by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/...
Source: chriszabriskie.com/preludes/
Artist: chriszabriskie.com/

[@nanamacapagal8342]

I had war flashbacks from grade 10 when a question like this showed up. "The triangle has sides 6, 8, and 10. Calculate the length of the inradius." Had to derive this formula from scratch, and I was RACING THE CLOCK to figure shit out

[@MathVisualProofs]

Glad you figured it out in time :)

[@11am2]

Area over semiperimeter

[@C00lestNerd]

Area is easy with right triangles like the 6 8 10. But I would have to derive the rest like you did.

[@logx-ow1us]

You mean 7th?

[@thejungwookim]

Did you just... use the area of a triangle to prove the area of a triangle?

[@Grizzly01]

No, he used one triangle area formula to prove a different triangle area formula.

[@thejungwookim]

@@Grizzly01 well yeah that's what I meant

[@sampathsris]

That's not even the best part. He introduced an unknown r in his final formula too. What a champ!

[@Grizzly01]

@@sampathsris You really don't understand this video, do you? 🤦🏻‍♂

[@phonemaster9306]

​@@sampathsrisThat r is the radius of the Circle

[@danielnidan]

use Heron's formula for area of triangle purely from the sides. Then to calculate r compare the 2 formulae. Heron's formula and this derived formula.

[@xninja2369]

umm I thought this was visual proof about herons formula but it was different

[@leonmareno4599]

hi @@xninja2369 Different but at the end there is the semi perimeter. And i still don 't know in heron s formula where is coming from.

[@logx-ow1us]

@@leonmareno4599 if you derive the formula algebraicly, you will find s. I did it earlier this year in 7th grade. Annoying as hell ngl

[@sketch4363]

Im so glad I just learned what an incenter is yesterday

[@Penguin-gm1fd]

Why couldn’t I have seen this BEFORE having a question on a test on almost the exact formula

[@Achill101]

Nice. It seems to be a good way to calculate the in-radius of a triangle.

[@frankreashore]

Very nice proof

[@kacper9385]

This idea has also extention into any n-sided form in which we re able to fit a circle (sorry for my bad English)

[@1973yogesh]

super

[@kevina5337]

I think the easiest and most intuitive way to prove A = ½bh Is to take any triangle, copy it and rotate it to get a rectangle/square/quadrilateral of some kind.. of course the area of this shape is just bh, recall that we duplicated the triangle to get this shape so you have to divide by 2. Voila!

[@sordidknifeparty]

Came here to say this. Now I don't have to

[@Int-E_]

That's really cool but I want to know how they actually figured this out or even thought that this could be smth coz I'm pretty sure that animation wouldn't be a valid proof

[@ajaybilare6627]

wooow... I haven't thought about it. great 👏👏

[@Beegeezy144]

Man, this one's good. 😳

[@one_logic]

Good thing it's not a Sierpinski triangle this time.

[@jacketofdiamonds]

...isn't A = bh/2 way easier to find than this? And way easier to prove?

[@MathVisualProofs]

Yes. This an area formula but it’s actually more useful for finding the inradius since you can find the area other ways.

[@jacketofdiamonds]

@@MathVisualProofs Oh, I get it now! Thanks.

[@MathVisualProofs]

@@jacketofdiamonds Perhaps I should have ended with solving explicitly for r...

[@mangaiyarkarasis644]

Can you do the perimeter of a triangle?

[@TruthOfZ0]

What is "r" equal to ??

[@bartsanders1553]

Are all triangles the sum of two right triangles that share a side of the same length?

[@MathVisualProofs]

Yes. You should be able to draw a perpendicular to the hypotenuse that passes through vertex opposite the hypotenuse right?

[@bartsanders1553]

@@MathVisualProofs I thought it seemed intuitive. And you know how math can be when you intuit things.

[@MathVisualProofs]

@@bartsanders1553 definitely. I haven’t checked every triangle :)

[@tomasbernardo5972]

Is this related to Heron's formula?

[@MathVisualProofs]

For sure it is!

[@TemmuzOcakoglu]

But how are we going to find r?

[@MathVisualProofs]

This is actually a theorem to help find r if you need it.

[@logx-ow1us]

@@MathVisualProofs I forgot the formula but u can go to art of problem solving intro to geometry chapter 6 and 7 to learn this really well.

[@HoSza1]

so how do you calculate r? this formula us useless only when you magically already know the inradius.

[@MathVisualProofs]

This is really about how to calculate r. So this is a formula relating values. If you know all but one you can find the other.

[@X-Studios.]

Why math gotta be this complex man 😢

[@kingo8914]

1 half base times height??

[@MathVisualProofs]

Yes. But this is a formula relating area to the inradius. So it is probably more useful in finding the inradius.

[@binarywizard69420]

.5 hb 💀

[@jamieelaalbarsha9302]

Greet What is program?

[@rahamathunisa1418]

Manim

[@vigneshbs5704]

Is a, b, c equal

[@MathVisualProofs]

No. This works for any triangle.

[@stefandempf818]

Verry Close to equal tho great eyesight

[@ravitasinghal5898]

Who’s going to tell you the R

[@MathVisualProofs]

This is actually more useful to find the R :)

[@snowdrop9810]

Best method is trig.

[@georgemaxwell3997]

Pointless video, as r is unknown and harder to find than the area

[@MathVisualProofs]

This isn't really about finding r - it is about finding a relationship between the area and r. So this could be useful for finding r.

[@georgemaxwell3997]

@@MathVisualProofs In this context, the sensible thing to do is to derive Heron's formula for the area of the triangle, as well as the formula for r, in terms of the semi-perimeter s. You only isolate an intermediate step. What does swinging out the triangles do for you? You can add up their areas just where they are. The danger in "graphical proofs", whether yours or Euclid's, is that it is easy to make unjustified assumptions. For instance, you just say "draw an incircle" and then mark right angles where the radii touch the sides. To see that this is true, you need some earlier theorems, which you do not make explicit. For example, someone might just say "draw the incircle of a quadrilateral" and then use some sort of graphical arguments to derive Heron's formula in that case. However, such a circle may not in fact exist and the formula is then false.