AWESOME Formula - AREA of a TRIANGLE (Herons Formula) 

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.

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

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).

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.

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.

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

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

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

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

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.

Area of shapes involving rectangle, square and triangles

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

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._

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!

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

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

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

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.

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.

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.

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

Thanks for sharing!

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!!! ;-)

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.

Good and clear presentation !

great and refreshing math lesson

Thank you so much

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

Thank you for the free Videos

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

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

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

very good. Thanks

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

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.

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

Outstanding video I am new subscriber to your channel

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.

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

Thanks!

Go right to the point

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

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.

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.

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

Great!

But where is the demonstration of the heron's formula

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

How about applying cosine law in trigonometry?

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

I fell asleep with the intro alone,.

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.

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

Thanks

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.

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

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

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

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

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

Awesome teaching,👍

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

A simple problem described in a complicated way

Nice video how to make math complicated.

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

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

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

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

amazing how could he have come up with the formula

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

Half the base multiply by altitude.

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. !

Interesting ❤

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.

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.

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.

Thanks you're and awsome teacher I loved it

Correction Should be square root of s(s-a) (s-b) (s-c)

Good lesson. Thanks.

Back up from area to calculate height... for fun...

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?

Great lesson

Did I get the area right?

Doubling the triangle into a rectangle = 20 cm /2 = 10

I could have said the same thing in 1 minute not 14! How did he come up with that formula. That's what I wanted to find out,

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

I got approximately 3.31 for the height and approximately 9.92 for the area...and if it was in centimeters, it would be approximately 9.92 square centimeters (not centimeters squared). C^2 is short for square centimeters...lol...also, learning obscure formulas is fun, but one can use simple tools to find the correct answer and not have to clutter their tool box with things that are hardly used and this "long way around" is not much longer than what you showed.

What a little beauty!

Quarter time of your lecture goes in introduction!!!

how about 4 times 5 =20 divide by 2 =10sq ft

But we can also use the heros fomular

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."

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

I did it in my head and got 10, so 0.1 out, that’ll do for me

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.

why not just ame a 4 x 5 rectangle and take half of that: 4x5=20/2=10.

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

I am special woman with special needs. Graduate from high school through special program designed to help people like me. I am still struggling with basic math like additions , subtraction....and so on. Do you have any program that can help me master these basic math