AWESOME Formula - AREA of a TRIANGLE (Herons Formula)
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- เผยแพร่เมื่อ 17 ก.พ. 2022
- TabletClass Math:
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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.com/
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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.
Thanks for sharing!
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
Nice.
Very clever
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
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."
I owe you an apology, sorry!
Good and clear presentation !
Thank you so much
very good. Thanks
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.
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
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.
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.
Same here.
It's even worse his intro is actually 5 minute exactly. What a jerk.
@@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 TH-cam employs. If so, there should be some sort of limitation for these people who purposefully create bloated content just to make money.
too much "plug" about himself!!😢
I could not agree more!
Just use cosine rule to get the an angle, then use ½abSin(C)
Thanks a lot John! Very useful formula!!! From Sardinia, Italy
Thanks!
Thank you for the free Videos
great and refreshing math lesson
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).
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.
@@footloose6382 You are absolutely right but that was not what my comment was about. Anyway.
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.
@@starfishsystems You are preaching to the choir.
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
Area of shapes involving rectangle, square and triangles
Go right to the point
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).
Thanks
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
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...
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
Measure the 2 diagonals and compare.
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.
A bullet-point presentation always works best for my super-charged neurons.
But waste of time
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
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._
That was my reaction. Look! I rote learnt a formula mum. This is not maths. Worse, this guy is teaching children.
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*
@@erikmarkus7467I mean... if you miss the point of a comment you can just post something completely off topic 🤷♂
@@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.
@@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.
You can use also cosine law by finding unknown height, then apply the formula of finding the ares of triangle..tnx
Which would be easier without a calculator?
But where is the demonstration of the heron's formula
Thank you so much. I might not be in your class but for some reason this helped me a lot so thank you
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.
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!
I got 9.9215674. Knotwilg got 9.92... Any comments?
Thanks, and it didn't take 14 minutes.
Got it!
@@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
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.
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.
Easy peasy!!!
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.
Great!
How about applying cosine law in trigonometry?
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!!! ;-)
iS NOT A RIGHT ANGLES TRIANGLE
Outstanding video
I am new subscriber to your channel
Took me back to school .Very clear,thank you Sir.Vetri South Africa 🙏🇿🇦🙏
I just did that in my head got to 9 in less than a minute.
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.
Incoherent.
You need to solve this long hand. I don't know how you got the answer!
After over 5 minutes of rambling, he finally gets to the problem.
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.
agree100%
"Focus is the key!" - Return to sender!
Easy, make two right Triangle and calculate the area as 1/2 x base and the height. Then multiply by 2
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.
the area of a 5x 4 rectangle would be 20 so the triangle is half of that therefore 10????
I fell asleep with the intro alone,.
I wish I could like your comment 1k times.
@@IbiRere Hello Mr Olakunie,thank you much,You are from Africa,Kenya i assume.
@@junpinedajr.8699 I'm from Nigeria.
@@IbiRere Thank you Mr Olakunle.😀😃😄😊
Since it is a right triangle, simply multiply te length times the width and divide by 2.
Interesting ❤
A simple problem described in a complicated way
You need to finish this by also asking what is the height, as this will require you to solve equation with 2 unknowns.
i need my popcorn listen your long explanation by solving this equation.
Good job. What software are you using to display the solution?
Thank you.
paint
Boy, oh boy, that Heron was one smart bird!
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.
Awesome teaching,👍
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.
What a little beauty!
Nice video how to make math complicated.
Idk about you but this helped me
1 mint. is sufficient for this question to solve but 15 mints. amazed***!
amazing
how could he have come up with the formula
You need to find the height first. Once you've found the height, use this formula
area = (base) (height)/2
The answer is 15/4•sqrt(7). Converting it to a decimal is only an approximation. That should be made clear.
Did I get the area right?
Great lesson
Half the base multiply by altitude.
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
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
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.
JHIll
How did you calculate the altitude?
@@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
The heron's formula is a must for students of math and geometry.
its even better when you derive it ...
since there is a 90 degree corner i fig it was half a square. therefoe 4 X 5 = 20 /2 = 10
But we can also use the heros fomular
I have a question
Sorry, I don't have the patience to wait for the solution and I was hoping to see the development of the formula.
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.
Yes, is important to take notes. But sometimes the problem start with the difficulty understanding notes.
Yes
how about 4 times 5 =20 divide by 2 =10sq ft
It's a right triangular...a picture worth a thousand words 🤔
Back up from area to calculate height... for fun...
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
A= 1/2 bh
bxh ÷2 = A
Good lesson. Thanks.
I've heard somewhere, "Show Your Work"! So, POINTS OFF & Unhappy Face.
9.447!
I was taught the proof of the Heron's formula in Hong Kong at Grade 9 in average Math class.
Ans Area = s(s-a)(s-b)(s-c) where s= (a+b+c)/2.
need sq root of that
Correction
Should be square root of s(s-a) (s-b) (s-c)
starts at 5:00
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.
!
Are you sure about that?
That only works for right triangles.
4+5+6=15 ÷4=3.75×3.75=14.0625
Thanks you're and awsome teacher I loved it
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.
Quarter time of your lecture goes in introduction!!!
Area=16.25 ans
Video starts at 7:30.
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.
Your information is great just TOO MUCH BABBLE!!!!
Pythagore and simple algebra give the same answer, Herons or not!
ooops. thought it was right angle at first..
Hahaha somebody fall asleep but not me i enjoy it. Tks
Start at 5.05....
by Heron' formula