Area of Triangle | Find out Height of Triangle | Half x Base x Height | Heron's Formula
ฝัง
- เผยแพร่เมื่อ 3 ส.ค. 2021
- Area of Triangle | Area of Triangle | Find out Height of Triangle | Half x Base x Height | Heron's Formula
One must watch this lecture. This method explained in the video is very helpful for making maths calculation easy specially in competitive exams.
Welcome to Nand Kishore Classes
For 8th, 9th & 10th (Mathematics)
New Batches start w.e.f. 1st April 2021 (Online)
To Fill the Registration Form, Click at below Link
forms.gle/FFRPDgMmTHq87MYu7...
Click at below links to download the PDFs containing TH-cam Links
1. Basic Math
nandkishoreclasses.com/basicma...
2. Shortcut Tricks & Reasoning
nandkishoreclasses.com/shorttr...
3. Kids Activities
nandkishoreclasses.com/kidsact...
4. Class 4th
nandkishoreclasses.com/class4...
5. Class 5th
nandkishoreclasses.com/class5...
6. Class 8th
nandkishoreclasses.com/class8...
7. Class 9th
nandkishoreclasses.com/class9...
8. Class 10th
nandkishoreclasses.com/class10...
9. Class 11th
nandkishoreclasses.com/class11...
10. Class 12th
nandkishoreclasses.com/class12....
Half x base x height
Two Methods to find out Area of Triangle | Half x Base x Height | Heron's Formula
Area of a Triangle from Sides
You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years.
It is called "Heron's Formula" after Hero of Alexandria (see below)
Just use this two step process:
Step 1: Calculate "s" (half of the triangles perimeter):
s = a+b+c 2
Step 2: Then calculate the Area:
herons formula A = sqrt( s(s-a)(s-b)(s-c) )
Example: What is the area of a triangle where every side is 5 long?
Heron's formula, also known as Hero's formula, is the formula to calculate triangle area given three triangle sides. It was first mentioned in Heron's book Metrica, written in ca. 60 AD, which was the collection of formulas for various objects surfaces and volumes calculation. The basic formulation is:
area = √(s * (s - a) * (s - b) * (s - c))
where s is the semiperimeter - half of triangle perimeter:
s = (a + b + c) / 2
However, other forms of this formula exist - if you don't want to calculate the semi perimeter by hand, you can use the formula with side lengths only:
area = 0.25 * √((a + b + c) * (-a + b + c) * (a - b + c) * (a + b - c))
Heron's formula proof
There are many ways to prove the Heron's area formula, but you need to know some geometry basics. You can use:
Algebra and the Pythagorean theorem;
Trigonometry and the law of cosines.
Other proofs also exist, but they are more complex or they use the laws which are not so popular (such as e.g. a trigonometric proof using the law of cotangents).
Algebraic proof
Triangle with sides a,b,c, height from right angle h, dividing hypotenuse c to segments d and c-d
In this proof, we need to use the formula for the area of a triangle:
area = (c * h) / 2
All the values in the formula should be expressed in terms of the triangle sides: c is a side so it meets the condition, but we don't know much about our height. So to derive the Heron's formula proof we need to find the h in terms of the sides.
From the Pythagorean theorem we know that:
h² + (c - d)² = a² and h² + d² = b², according to the figure above
Subtracting those two equations gives us:
c² - 2 * c * d = a² - b² from which you can derive the formula for d in terms of the sides of the triangle:
d = (-a² + b² + c²) / (2 * c)
Next step is to find the height in terms of triangle sides. Use the Pythagorean theorem again:
h² = b² - d²
h² = b² - ((-a² + b² + c²) / (2 * c))² - it's already in terms of the sides, but let's try to reduce it to nicer form, applying the difference of squares identity:
h² = ((2 * b * c) - a² + b² + c²) * ((2 * b * c) - a² + b² + c²) / (4 * c⁴)
h² = ((b + c)² - a²) * (a² - (b - c)²) / (4 * c²)
h² = (b + c - a) * (b + c + a) * (a + b - c) * (a - b + c) / (4 * c²)
Apply this formula to first equation, the one for triangle area:
area = (c * h) / 2 = 0.5 * c * h
area = 0.5 * c * √((b + c - a) * (b + c + a) * (a + b - c) * (a - b + c) / (4 * c²))
area = 0.25 * √((b + c - a) * (b + c + a) * (a + b - c) * (a - b + c)
Here you are! That's the Heron's area proof. Changing the final equation into the form using semiperimeter is a trivial task.
Trigonometric proof
Triangle ABC with sides a,b,c and angles α β γ
Have a look at the picture - a, b, c are the sides of the triangle and α, β, γ are the angles opposing these sides. To find the proof of Heron's formula with trigonometry, we need to use another triangle area formula - given two sides and angle between them:
area = 0.5 * a * b * sin(γ)
Welcome to Nand Kishore Classes
Facebook Page -
/ nandkishorec. .
TH-cam Channel -
/ nandkishore. .
Instagram -
/ nandkishore. .
Twitter -
/ nandclasses
Website
nandkishoreclasses.com
Thanks 👍👍👍👍👍👍👍
Welcome
"Mathe-Re-Chul".
Heron's formula, formula credited to Heron of Alexandria (c. 62 ce) for finding the area of a triangle in terms of the lengths of its sides. In symbols, if a, b, and c are the lengths of the sides: Area = Square root of√s(s - a)(s - b)(s - c) where s is half the perimeter, or (a + b + c)/2.
Heron's formula is a formula for calculating the area of a triangle in terms of the lengths of its sides that is credited to Heron of Alexandria (c. 62 CE). If the lengths of the sides are a, b, and c in symbols, then: A = √{(s - a)(s - b)(s - c)} , where s is half the perimeter, or (a + b + c)/2.
In Heron's formula 's' stands for semi perimeter. Q. The area of a triangle, with sides a, b, c and semi-perimeter s, is: Q.
Heron's formula can be applied to any type of triangle. Since, Heron's formula is used to calculate the area of a triangle and every type of triangle in this world will have some area. So, for a triangle, if the three sides are known, you can directly calculate the area using Heron's formula.
Heron's Formula was discovered by Heron of Alexandria (also known as Hero of Alexandria) who was a Greek Engineer and Mathematician. He found the area of the triangle using only the lengths of its sides which made it possible to apply to any type of triangle be it, equilateral, isosceles, or scalene.
Heron of Alexandria (flourished c. ad 62, Alexandria, Egypt) was a Greek geometer and inventor whose writings preserved for posterity a knowledge of the mathematics and engineering of Babylonia, ancient Egypt, and the Greco-Roman world. Heron's most important geometric work, Metrica, was lost until 1896.
Heron's formula computes the area of a triangle given the length of each side. If you have a very thin triangle, one where two of the sides approximately equal s and the third side is much shorter, a direct implementation Heron's formula may not be accurate.
The formula for area:
Area Formulas
Area of a rectangle is the length times the width. Area of a parallelogram is base times the height. Area of a trapezoid is one half the sum of the two bases times the height. Area of a circle is π times the square of the radius.
( kmadhavameducation@gmail.com / +91-8252771261).
this video have cleared my ch-12 of maths of 9th class chaper name is same Heron's formula 🤩
Me in class 8th 😐😐😐
Thank you so mach Sir
Big fan sir and tq for solution💗🙏💖🙏🙏
Thankyou sir very helpfull video.
Thank you so much sir
Welcome
Nice Video Sir 🙏🏼
It Will Help In My Upcoming Exams 💐
All the best
Thanku Sir
Thank-you respectful sir
Welcome
one of best lecture of geometric. especially Herone formula
Thanks
Sir thanks a lots bahut hi nice explanation andaj me bataya
Thanks
It was so good to learn from you, even if I am a 5th std boy still I understood it with such a good formula. Thanks a lot 🙏 💓
God bless you
Best knowledge in shot video thankyou sir
Welcome
Thank you , it was really easy to understand.
Welcome
@@NandKishoreClasses sir can you please help me
I am preparing for JEE
Sir can you please explain how to find height of scalene triangle because you said that herons formula is best so there will be another way?
Sir please help me
Easy to understand
Very good sir. Thanks.
Thanks
Thank sir
Welcome
Very helpful video ❤
Thanks
Thanks ❤❤❤❤
Good Video 😊😊
Thanks
Thanks sir
Sir, u r my world.
Thanks
Thanks for this video
Welcome
Nice video sir 💕
How to solve if c is not given and s is given
Well explained 🙏 thank you 👍
Welcome
Well explained. Keep it up .
Thanks
Thanks for explaining
Welcome
3*3✓3=?
You teach very good and I also understand
God bless you
Teaching classes so best
Thanks
Thanks sir:)
Welcome
Thx sir its my final exam i watch you first time but now i will watch you regularly
All the best
Wery well explained thanks sir 🙏
Thanks
Very nice sir ji
Thanks
Very nice sir mera Mind me aa gaya hai
Really helpful
Thanks
Awesome sir
Thx
Nice Sir. Chandrima
Thx
If perpendicular not bisects base, still formula gives good results true or false
Plz give me your free fire uid our name
Pleas galdi
True
❤❤
🙏sir kya gajab ki teaching hai. Brilliant aap ka contact number plz. Mujhe bahut accha laga. Coaching chahiye
Thanks for writing. Whatsapp at 6283505240
Fail
👍👍👍👍👍👍
Thanks
Nice
Thanks
square Foot mein kesy nikalty hain ye centimeter hay
Sapana😊
~hello~
Sir please at triangle jiski side jo hai 13 cm 14 cm of 15 CM hai uska height kaise nikaalenge please bata do
2nd method use krp
Missing number of addition and subtraction for class 6
Sarkar hamare pass 17 ,10 and 9 sides vah To hamara kya answer
Sir 9 kaise aya
Sir 25-9 kue less kiya
😢
A city is 10 km long and 7km wide. Calculate it's area. Ans
Area of rectangle=length *breadth
=10km*7km
=70km²
Uncle piya ho kua,
Thank you but I don't getting
😂
Cm ma squre kyu plaga😢
•.•
السلام علیکم بھائی یہ زمین ٹوٹل کتنے مرلہ ہوئی...خیبر پختونخواکے ضلع چارسدہ کے ایک گاؤں ہے....شمال 53.75== مشرق 31.75===جنوب 53.75==مغرب... 34 == یہ ٹوٹل کتنے مرلے کا پلاٹ بنتا ہے...
By herons formula to aapka answer 12cm aur jab simple formula yani sum of all side to answer 16 cm aa raha hai
Bilkul samagh mien nahi aya
Tumare pass dimak nhi he
I don't understand
😂 koi nhi smj jaoge
Baba black sheep
I don't understand 2method 6:27
If you dont know maths then you will never understand it
@@dibyasorupadas1678 You have understand
Sala pua 😂😂😂😂
I don't understand 😢😢😢😢😢😢😢😢😢
Thank you so much sir