AWESOME Formula - AREA of a TRIANGLE (Herons Formula)

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Math help with finding the area of a triangle using Herons formula. For more math help to include math lessons, practice problems and math tutorials check out my full math help program at tcmathacademy....
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[@davidbrisbane7206]

Name the vertices of the triangle as follows. Botom left is A, top is B and bottom right is C The area of the triangle is (1/2)*c*b*sin(A) =(1/2)*4*6*sin(A). By the cosine rule 5² = 4² + 6² - 2*4*6*cos(A) ⇒ cos(A) = (4² + 6² -5²)/(2"4*6) = 27/48 ⇒ A = 55.7711° (4 decimal placed) ⇒ sin(A) = sin(55.7711°) = 0.8268 (4 decimal placed) So, Area of ΔABC = (1/2)*4*6*sin(A) = (1/2)*4*6*0.8268 = 9.9216. Interestingly, this approach can be used to prove Heron's formula. If instead of computing the area with numbers we use the side length of a, b, c, then we could have found sin(A) from cos(A) using the fact that sin²(A) + cos²(A) = 1, so sin(A) = √(1 - cos²(A)). We let s = (a + b + c)/2 and then we apply some algebra on s, a, b & c in the equation, noting how to factorise the difference between squares (twice) to find sin(A). From there it is an easy step to find the area of the triangle using area = (1/2)*c*b*sin(A) and we end up with Heron's formula.

[@ShanDysigns]

The 'quick' introduction lasted 3 minutes and 48 seconds, that's 26.54% of the whole video. I don't understand these long, meandering introductions on your videos. I love watching math techniques, I just find these marathon intros to be pointless.

[@denisenoe7746]

Same here.

[@lapulapu7422]

It's even worse his intro is actually 5 minute exactly. What a jerk.

[@ShanDysigns]

@@lapulapu7422 I'm beginning to think that he is pushing these videos to a certain minimum time for the sake of how many commercial breaks KZbin employs. If so, there should be some sort of limitation for these people who purposefully create bloated content just to make money.

[@alexanderpo185]

too much "plug" about himself!!😢

[@wolfgangsuehrer5405]

I could not agree more!

[@danmyers2338]

Not knowing Heron’s Formula, I went the long way around the barn. I drew in the altitude as you did. Now I have two triangles with a common height and bases of X and 6-X. I used the Pythagorean Theorem to write the equations for h^2 of each triangle, set them equal to each other and solved for X. Using that information, I calculated the areas of the two triangles, added them together and got the same 9.92 answer, it just took a little longer (maybe). FYI, h= 3.31 and X=3.75

[@johnhebert3855]

Nice.

[@bravemind8152]

Very clever

[@mraoufmd]

That’s exactly what I did and it produces the result The base which is 6 cm is divided between the two triangles to 2.25 and 3.75 Height is sq. Root of 10.9375 = 3.307189 Area is 1/2 of 6x 3.307189 = 9.9215

[@quabledistocficklepo3597]

Dan Myers and mraoufmd What? "TableClass Math" talked too much, but you didn't talk enough, for me, at any rate. How about adding a few steps? I took Algebra in High School--successfully-- about 65 years ago, but I don't see how you "solved for x."

[@markmorley7938]

I owe you an apology, sorry!

[@knotwilg3596]

Heron's formula actually gives the simplified solution 15/4 * sqrt 7 = 9.92... All the short cuts mentioned in the comment section start from the illusion that the upper angle is 90 degrees. The drawing is deceptive and the angle is close, about 83 degrees. So if you double the triangle it won't give a new triangle because the left leg won't continue in a straight line (180 degrees).

[@footloose6382]

There is no need for that angle to be a right angle or any other specific angle. Any triangle can be translated to a parallelogram by taking an exact copy any placing the two hypotenuses together, the area is then the length x the height divided by two.

[@knotwilg3596]

@@footloose6382 You are absolutely right but that was not what my comment was about. Anyway.

[@starfishsystems]

There is no.illusion. You just made a bad, and easily avoided, assumption. Unless the problem is explicitly stated or depicted as involving a right triangle, the correct assumption is that it's instead the general case. In this particular example, we can also quickly verify that it's NOT a right triangle because the squares of the sides don't sum.

[@knotwilg3596]

@@starfishsystems You are preaching to the choir.

[@boeingnz]

Yeah it's not a right angle triangle. a^2 + b^2 not equal to c^2. The mirror triangle to get a rectagle seem to work too. I've been taught that way but all forgotten now. LOL

[@shujaathusain6738]

Find one of the angles by using the cosine rule. Then get the area using the sine rule: Area = 0.5 a x b x sinc

[@maxpercer7119]

I wouldn't call that 'using the sine rule' , since you are just using the definition of sin C. In my mind there is a distinction between rules and definitions, but they are often blurred. The sine rule, analogous to the cosine rule, is also known as the rule of sines or law of sines is a way to method to 'solve' a triangle and can be expressed by the equation a / sin A = b / sin B = c / sin B . Fun problem, convert sin C to an algebraic expression. Hint, sin (arccos (x)) = √ 1 - x^2. I think it comes out to be equivalent to heron's area formula, after some algebra. Also to nitpick further that should be , Area = 0.5 a x b x sin C , where angle C is opposite side c , notice the caps on C.

[@aidenokowa9982]

Go right to the point

[@Dave-zu6sm]

Just use cosine rule to get the an angle, then use ½abSin(C)

[@christopherbedford9897]

In any Math exam I ever took, writing down a formula for the desired result and then calculating it would get you one point for knowing how to apply it to the specific case, and maybe a second point for doing the arithmetic correctly, out of a possible max of probably 4 or 5 points -- *IF* the formula is taught in the curriculum. If it _isn't,_ (and I never learned this one in 12 years of school and three of university) then writing down this formula and calculating the answer would probably get you no marks at all because the whole point of Math tests is not to come up with an answer like a History exam but to _work it out._

[@benchapple1583]

That was my reaction. Look! I rote learnt a formula mum. This is not maths. Worse, this guy is teaching children.

[@erikmarkus7467]

i mean... if you don't want to use herons formula you can just use pythagoras theorem and figure out the height of the triangle *shrug*

[@christopherbedford9897]

@@erikmarkus7467I mean... if you miss the point of a comment you can just post something completely off topic 🤷‍♂

[@erikmarkus7467]

@@christopherbedford9897 I have no idea what your problem is for writing down this formula and using it to calculate the area of a triangle of known sides - that is what the formula is for. I learned it when I was a freshman at highschool. There's not much to "work out" when the problem is stated as 'calculate the area of a triangle with these side lengths'. Just like given a problem of calculating the area with side a and it's altitude. You just write down a*h/2 and punch in the numbers. You don't actually prove the formula every time by completing a rectangle and taking it's half. Why would you need to prove or work out Heron's formula? You just use it because it's elementary... IF however you don't like using it, then just use Pythagoras theorem to calculate the h first out of the given three sides. I imagine you still wouldn't include a proof of the theorem, just go ahead and use it.

[@christopherbedford9897]

@@erikmarkus7467 The "problem" is very clearly stated in my original post: Mathematics is not a rote learning subject, it's logic and working it out subject. If you want to learn formulas and rattle them off, good for you, but you'd have been better off studying a subject like History or Law. But even then, the idea behind learning those subjects is to understand the world around you abd apply your knowledge of one situation to analyse a different one. If you think the object of Maths is to find a number that represents the area of a geometric shape your education failed you sadly.

[@junpinedajr.8699]

I fell asleep with the intro alone,.

[@IbiRere]

I wish I could like your comment 1k times.

[@junpinedajr.8699]

@@IbiRere Hello Mr Olakunie,thank you much,You are from Africa,Kenya i assume.

[@IbiRere]

@@junpinedajr.8699 I'm from Nigeria.

[@junpinedajr.8699]

@@IbiRere Thank you Mr Olakunle.😀😃😄😊

[@tonywright560]

Like Dan Myers, I started out with the Pythagorean Theorem and tried to use what I call substitution of equations to find the height. I was too lazy to work it all out but my gut feeling is that by using that method, you might wind up deriving Heron's Formula. I'm not sure of that. It's just intuition. But at the end of the day, I thank Heron and I thank you for telling me about that formula.

[@timspooner59]

My maths teacher 12 to 18 was fantastic. Home work every night was a published maths test of 90 minutes.99% of my cohort passed O and A levels (UK).

[@johndonne7807]

My wife and have 4 advanced degrees between us including PhD’s. I have a professional license to teach 7-12. My mentor was teacher of the year in America. I’ve taught mathematics mastics at virtually every level up to entry level graduate studies. This man represents everything that’s wrong with math education. He droned on for more than half the video without saying a single relevant word. He then distracts the student with useless references to law of sines/cosines which at this level would be utterly opaque to the students. Then “magically “ a formula appears. Absolutely no intuition motivating it, and absolutely no mathematical reasoning as to how this formula is derived. He then instructs students to blindly apply this formula, and does a poor job of that as well. Moreover, he speaks for 15 minutes of which around 3 are actually relevant to the problem. To believe that this drivel is within the attention span of a teen student is delusional. This isn’t remotely on the spectrum of modern mathematics education. This could be used as an example of what is wrong in math education in this country. Somebody stop him before he disenfranchises another child from learning to reason mathematically.

[@gn2932]

I agree completely!!! Most is drivel as you said.

[@jimlocke9320]

I agree! Tabletclass Math should, at the minimum, compute the height of the triangle, which is a common side to 2 right triangles with hypotenuses a and b, and sides, let's call them p and q, where p + q = c. He can readily eliminate p and q to generate a formula for the area in terms of a, b, and c. With a fair amount of algebra, that formula will simplify to Heron's formula. He could show the algebra without going through the steps.

[@lederhudler]

This guy should get a job at Walmart or something. He should not be teaching math anywhere. The formula is clever, although it could be derived using algebra and geometry. I wish I had heard of it before. But to drone on and on about nothing. Yikes I'd have fallen asleep in his class.

[@ayeflippum]

*john donne* I agree. I played the video at x2 speed and it *still* took forever for him to get to the point.

[@bobkod9415]

Agree. He is repelling rather than attracting interest. But he is excellent in self-advertising of his own "achievements".

[@StephenSelelya]

Good and clear presentation !

[@brianwalters9768]

Showing students how to use a formula is not teaching mathematics. Problem solving and formula derivation is always the best approach. It will develop a true understanding.

[@julietmoraa-te6nu]

Area of shapes involving rectangle, square and triangles

[@olivierhabineza7755]

First you find the half of 4+5+6=15 that is 7.5. Then you find the square root of the product 7.5(7.5-6)(7.5-5)(7.5-4)=7.5*1.5*2.5*3.5=98.4375, that is 9.9215.... square units. Thank you!

[@4DMovie]

I got 9.9215674. Knotwilg got 9.92... Any comments?

[@snapcutter9596]

Thanks, and it didn't take 14 minutes. Got it!

[@davidbroadfoot1864]

@@4DMovie Your answer is not really correct. Knotwig is correct. If you had said, "I got 9.9215674..." or if you had said, "I got 9.9215674 correct to seven decimal places" you would've also been correct. I got 9.92156741649221 correct to 14 decimal places

[@quabledistocficklepo3597]

Olivier Habineza a+b+c/2=height? Since when? Where did you get that formula? If you use it on a 3X4X5 triangle you get 12/2=6, which is Not anything of any value.

[@terryjohinke8065]

I used to teach Heron's Formula, along with pythagos' Teorem and trigonometry. We'd use markers of three types as mentioned and students assignment was to calculate an area , on the football oval , to use all these formulas. Yhat was year 11, here that"s pre last year of high school. Good fun. Lots of number crunching if the problem is a fraction or decimal. So we set up problems that s added to an equal number, dicided by 2 to get s. Easier to teach if it's not as cumbersome as this one.

[@joekelley8084]

I flipped the top down and made a rectangle but also had to flip sideways so that it was a perfect rectangle 4 by 5. Makes 20 square and had to half it making the correct answer 10 square.

[@larryparish8708]

Easy peasy!!!

[@jmack619]

That's what I thought at first. Works for a perfect right angle... ex 3,4,5 ... 3sq +4sq=5sq. But this is not quite à right angle, I guess.

[@everyonehasadream]

A bullet-point presentation always works best for my super-charged neurons.

[@vishvanathkoti3808]

But waste of time

[@LordWilsonVILLA]

I got the answer to 10 in a lot simpler way = 4x5 = 20 which is the area of a square of 4 by 5 and halve it to get the area of the triangle = 10 - simple!!!! To find the area as described in this video is why so many students get so bored with maths YES MATHS!!! The abbreviation of mathematicS is MATHS!!! ;-)

[@careyharborne8357]

iS NOT A RIGHT ANGLES TRIANGLE

[@Brettly01]

4 5 6 means 4 and 5 are at 90degrees Draw a line opposite 4 and 5 at the same measurements. Now you have a 4x5 rectangle, 4X5=20 half of 20 is 10 (cos you want to know the area of the triangle which is half the rectangle) The answer is 10

[@petrmatejka4900]

You mistaken it with 3-4-5 triangle - which is right triangle. 4-5-6 is NOT right triangle: quick check 4^2+5^2 = 41 not equal to 6^2=36. so Pythagoras theorem is not working here...

[@marcellofanutza3387]

Thanks a lot John! Very useful formula!!! From Sardinia, Italy

[@bertrandviollet8293]

But where is the demonstration of the heron's formula

[@lolitotorlao7148]

You can use also cosine law by finding unknown height, then apply the formula of finding the ares of triangle..tnx

[@trailertrish2587]

Which would be easier without a calculator?

[@waltervetri2476]

Took me back to school .Very clear,thank you Sir.Vetri South Africa 🙏🇿🇦🙏

[@virgiliobartolo5725]

Nice formula but you didn't show how this formula was derived like what is "s" referring to. You just used the formula and plugged the numbers and I don't see mathematical justifications as to where the formula came from. But thank you and I will appreciate it if you can prove the validity of the formula.

[@ronrobinson3872]

agree100%

[@duggydo]

After over 5 minutes of rambling, he finally gets to the problem.

[@cgunnikrishnan79]

A simple problem described in a complicated way

[@baxbanni2226]

"Focus is the key!" - Return to sender!

[@Fogclan]

The simplest way would Be to take another triangle of the same dimensions and rotate it and join it along the current base of 6. This results in a rectangle with a L= 5, W=4. L x w = 20 and half of that = Area of 10 for the original triangle.

[@markg2851]

i need my popcorn listen your long explanation by solving this equation.

[@ourv9603]

Finding the area of a triangle is easy. Double it. Put 2 triangles side X side makes a rectangle. Calc the area of the rectangle & divide by 2. DONE. !

[@marcelstrydom3200]

Are you sure about that?

[@jamesbrown8766]

That only works for right triangles.

[@juliesoal1207]

the area of a 5x 4 rectangle would be 20 so the triangle is half of that therefore 10????

[@amandalee714]

Thank you for the free Videos

[@henrys9244]

I just did that in my head got to 9 in less than a minute.

[@mustafaoguzmizrak8792]

It's very easy with the Formel of Heron. The area of a triangle iş: A=[u*(u-a)*(u-b)*(u-c)]^½ --> u= (a+b+c)/2

[@Cladman3001]

Easy, make two right Triangle and calculate the area as 1/2 x base and the height. Then multiply by 2

[@iamlearningalifbata]

Outstanding video I am new subscriber to your channel

[@maximatias4146]

great and refreshing math lesson

[@dr.echevers]

Nice video how to make math complicated.

[@tylerbeckett4560]

Idk about you but this helped me

[@dannypiercesr]

Hey John , thank you for all the tutoring, I really enjoy you videos and learn from them . I have a question , how to figure the cross section of a rectangle, to check for square

[@alext8828]

Measure the 2 diagonals and compare.

[@a_j6650]

1 mint. is sufficient for this question to solve but 15 mints. amazed***!

[@jimivey6462]

Boy, oh boy, that Heron was one smart bird!

[@siavashghazisaidi8338]

The heron's formula is a must for students of math and geometry.

[@ronrobinson3872]

its even better when you derive it ...

[@tylerbeckett4560]

Thank you so much. I might not be in your class but for some reason this helped me a lot so thank you

[@ralph1478]

Since it is a right triangle, simply multiply te length times the width and divide by 2.

[@alexcampbell1895]

Thanks for sharing!

[@jamesmoore1751]

I saw it as half of a rectangle of length 5 and height 4; so the rectangle's area is (5 x 4) = 20, so the triangle is half of that, or 10.

[@jerrysherman2743]

I thought the exact same thing, but apparently, that only works if the angle at the top is 90 degrees, and that cannot be true if the sides are exactly the lengths shown.

[@fedup3449]

@@jerrysherman2743 Yes you can be sure that the angle is 90. In a r angled triangle, c (in this case 6) = the sq. root of the sum of the squares on the other 2 sides. So, (4x4 + 5x5) = 42, so the sq. is 6.48 which is 6 in building terms or in any country where profit is more important than accuracy. As such, the area is 10, half of the rectangle that is 4x5.

[@jerrysherman2743]

@@fedup3449 In math, accuracy is important, regardless of the country. In this case, the square of the triange in the formula is 9.9215674. Plugging that value into the formula to find the height, the height comes to 3.307189.

[@julietdelacruz502]

How about applying cosine law in trigonometry?

[@michaelharris8554]

Total the three sides divide by 2 , call the result s. Subtract the respectively sides from S. Múltiples all 3 value, then find sq rt of product. Múltiples s by sq rt of earlier product.

[@christopherbedford9897]

Incoherent.

[@michaelsteinkugel8174]

Its easy. Use the theorem of euclid and you have immediatly the high. Wen you have the high of the triangle, you can calculate the area (a x b : 2).

[@hulamei3117]

You need to solve this long hand. I don't know how you got the answer!

[@kasperkuikka5279]

Thank you so much

[@njd2342]

Looks like a right angled triangle so (4x5)/2 = 10. Can't be given Pythogoras (as hypotenuse would be 6.4 not 6) Your exact measure of 9.9 ain't too far away from 10.

[@bighands49]

The answer is 15/4•sqrt(7). Converting it to a decimal is only an approximation. That should be made clear.

[@c-channel800]

very good. Thanks

[@vanessawelles4760]

You need to finish this by also asking what is the height, as this will require you to solve equation with 2 unknowns.

[@joycemutare8635]

Thanks

[@lawrencedavis2089]

It's a right triangular...a picture worth a thousand words 🤔

[@ItsTonez]

Can't i just divide the 6 to 2 and then make the triangle 2 right triangles. Then use Phythagorean Theorem. Then use the formula 1/2 • b • h? Will the results be the same?

[@jonathanau8735]

There is a formula, square root of s minus a, multiplied by sminus b, multiplied by s minus c, where s is the sum of a plus b plus c.

[@gapster77]

If this is a right angled triangle, then imagine putting two of the same triangle together, hypotenuse to hypotenuse, now you have a square 4 x 5, giving an area of 20. So just the one Triangle would be half of this area …so the area of the Triangle is 20/2 = 10. Tri’ Area = (Adj’ x Opp’)/2

[@lk-wr2yn]

I took your formula also first, than I checked it by this : 25-x^2=16-(6-x)^2 than h=(25-3.75^2)^0.5

[@duggydugg3937]

amazing how could he have come up with the formula

[@enosmutowelasiwakwi1859]

Thanks!

[@f.osborn1579]

What a little beauty!

[@白露时光]

I swear i had an interest in knowing how TCM was going to work it out but I did not know when I fell asleep, completely, and in broad daylight.

[@canman5060]

I was taught the proof of the Heron's formula in Hong Kong at Grade 9 in average Math class.

[@kevinspencer9245]

axa + bxb = cxc works for 90 degree triangle. 4x4 + 5x5 = 6x6 16 + 25 = 36, true So, use legs 4 and 5 for base and height. 4x5/2 = 20/2 = 10 square units

[@MrRod4000]

16 + 25 = 41, NOT 36.

[@kevinspencer9245]

Thanks. Went too quickly. Thinking of a 3-4-5 and didn't do the math. I'll return to prob

[@michaelempeigne3519]

false. 16 + 25 = 41 furthermore, if you want to be able to tell if a triangle is acute, obtuse or right angled with no trigonometry. It is possible to do that.

[@humanforotherhumans]

Yes, is important to take notes. But sometimes the problem start with the difficulty understanding notes.

[@vishvanathkoti3808]

Yes

[@lowellmccormick6991]

I guess it's cheating but, I opened AutoCAD, drew a line 6" long. Draw a 4" radius circle with center at the left end of 6" line. Draw a 5" circle with center at the right end of 6" line. Draw lines from each end of 6" line to the intersection of the circles. Turn the lines into polylines and join them. List the triangle and read 9.92 sq inches from the list.

[@cyruschang1904]

You need to find the height first. Once you've found the height, use this formula area = (base) (height)/2

[@hassegreiner9675]

Sorry, I don't have the patience to wait for the solution and I was hoping to see the development of the formula.

[@RockyTremblay]

Good job. What software are you using to display the solution? Thank you.

[@careyharborne8357]

paint

[@krisainsworth239]

Your information is great just TOO MUCH BABBLE!!!!

[@awandrew11]

Pythagore and simple algebra give the same answer, Herons or not!

[@jhill4874]

Not knowing Heron's formula, I did this the hard way. I calculated h of the triangle (3.3) and performed the regular A=(1/2)bh.

[@quabledistocficklepo3597]

JHIll How did you calculate the altitude?

[@jhill4874]

@@quabledistocficklepo3597 Shorthand: ()2 means the contents of the parans is squared. (x)2 + (h)2 = (4)2 = 16 (h)2 = 16 - (x)2 (6-x)2 + (h)2 = (5)2 = 25 (h)2 = 25 - (6-x)2 (h)2 = 25 - 36 + 12x - (x)2 (h)2 = 12x - 11 - (x)2 16 - (x)2 = 12x - 11 - (x)2 16 = 12x - 11 27 = 12x x = 27/12 (h)2 = 16 - (27/12)2 = 16 - 729/144 = 2304/144 - 729/144 = 1575/144 h = 39.68/12 = 3.31

[@samuelkelly2197]

U.K. That was really a fun way to solve the problem. Some comments were highly ridiculous but please overlook them. Those comments are from those who will usually complain about their Maths Tutors as boring. They are those who think they are smarter than Archimedes. Maths should be fun and you made it look so. I have watched a few of your clips and I am yet to give a negative comment. You are a 5 🌟Tutor. I am considering acquiring your Algebra and Geometry Notes to help my boys at home. Looking forward to your Calculus Notes. Cheers

[@dey2316]

I can see why pupils in this teachers class would be bored out of their brains within a very short time. It was sooo drawn out. And I wouldn’t pay much attention to a math teacher who doesn’t know how to pronounce Pythagoras.

[@scottsamoly2404]

since there is a 90 degree corner i fig it was half a square. therefoe 4 X 5 = 20 /2 = 10

[@debasisdatta6337]

Why call it as Heron's formula? This is an adoption of the most elegant formula by Maharshi Brahma Gupta for calculating the area of a cyclic quadrilateral.

[@davidbrisbane7206]

The equivalent formula for a cyclic quadrilateral (i.e. a four sided shape whose vertices lie on an inscribed circle) is area of ABCD = √[(s - a)(s - b)(s - c)(s - d)], where s = (a + b + c + d)/2. This is Brahaguta's formula. *Note:* if the length of side d is shrunk down to zero, then the four sided figure becomes a three sided figure, I.e. a triangle and the above formula reduces to area =√[(s - a)(s - b)(s - c)(s - 0)], where s = (a + b + c + 0)/2, which is area = √[s(s - a)(s - b)(s - c)], where s = (a + b + c)/2. I.e. Heron's formula. Although this reduction to Heron's formula appears to be only valid for a circumscribed triangle (I.e. a triangle whose vertices on lie on a circle), this formula is true for all triangles, as every triangle can be circumscribed by a circle.

[@walterbeaver6365]

You talk too much. "KISS" statement. "Keep It Simple Stupid" Very effective resource for teaching Students loose interest by different means.

[@t.s.srinivasaraghavan7387]

Half the base multiply by altitude.

[@edsolano1162]

4+5+6=15 ÷4=3.75×3.75=14.0625

[@AnikaHaywood]

Interesting ❤

[@vraagal]

When I look closely, I see a rectangular triangle. Why? 4^2+5^2=6^2 (Phytagoras). The area of this recangular triangle is therefore (4*5)/2=10.

[@kevinbranshaw7294]

Wrong. It's not a right triangle.

[@jonathanfrayne9501]

@@kevinbranshaw7294 Er, RIGHT! a 4,5,6 triangle IS a right angle triangle.

[@vraagal]

@@kevinbranshaw7294 You are right. I made a miscalculation. 16+25=41 not 36. Blame on me 🙂

[@careyharborne8357]

@@jonathanfrayne9501 3, 4, 5 is a right angles triangle, not yours.

[@pareshmehta4670]

Quarter time of your lecture goes in introduction!!!

[@larrypredika8744]

Think outside the box. Extend parallel lines of 4 and 5 . 4x 5 is 20 / 2 = 10 even!

[@rsmickeymooproductions4877]

That's how I did it. Took me 5 seconds

[@niranjanchakraborty1139]

Ans Area = s(s-a)(s-b)(s-c) where s= (a+b+c)/2.

[@lylestavast7652]

need sq root of that

[@douglaseLamb]

9.447!

[@bahaah5218]

There is faster method to solve this problem: a=4, b=6, c=5 AxC/B=h , h= 4x5/6= 10/3 Area = 1/2x base x h= 0.5 x 6 x10/3= 10 Ans: 10

[@anniesworldeverythingsurro8214]

9.9215

[@saidrabie5081]

Great lesson

[@jomaragosita6518]

Hahaha somebody fall asleep but not me i enjoy it. Tks

[@hezronmbati96]

But we can also use the heros fomular

[@isaacmunroe9159]

Awesome teaching,👍

[@JT-nf9tk]

I just made the triangle into a square so that the area of the square is 20. Since the original triangle is half the square then the area is 10. Figured it out in 5 sec.

[@mikesanders4012]

Maybe SAVE all of the personal info for last, for those interested & all.

[@BernardGreenberg]

I guess he wasn't called "Hero" for nothing (j/k). Far more interesting would be to draw the inscribed circle and six triangles that result and show how latter-day heroes and heroines can arrive at his wonderful result. In the old days, we had books of such magic formulae (even from antiquity), but understanding their derivation "teaches one to fish for him or herself."