Heron's formula provides a way to calculate the area of a triangle using only the lengths of its three sides. It was developed around 60 AD by Heron of Alexandria, a Greek mathematician. The formula uses the semi-perimeter of the triangle and the differences between the semi-perimeter and each side length. While Archimedes may have known of the formula earlier, Heron was the first to provide a clear written explanation. Over time, the formula has taken on different mathematical forms but provides the same calculation of the triangle's area based on its three side lengths.
One of the best PPT on HERONS' FORMULA You will get here.Contains all most all information about Heron, its formula.Formulas of some other shapes also.Area of triangles and its derivation.
I can say this power point will help you to know about "HERON’S FORMULA". You will understand the meaning of 'HERON’S FORMULA'.This power point is best for class 9.
Monish Jeswani.
Thank You!!!!
One of the best PPT on HERONS' FORMULA You will get here.Contains all most all information about Heron, its formula.Formulas of some other shapes also.Area of triangles and its derivation.
I can say this power point will help you to know about "HERON’S FORMULA". You will understand the meaning of 'HERON’S FORMULA'.This power point is best for class 9.
Monish Jeswani.
Thank You!!!!
this is about surface area and volume to help the students to do there projects or ppts and insure that u can also see this and make another like this so all the best of this ppt for who al cannot do on there own so enjoy this thing here .... and thanks for watching :) ..
this is about surface area and volume to help the students to do there projects or ppts and insure that u can also see this and make another like this so all the best of this ppt for who al cannot do on there own so enjoy this thing here .... and thanks for watching :) ..
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
3. Heron's formula is named after Hero of Alexandria, a
Greek Engineer and Mathematician in 10 - 70 AD.
You can use this formula to find the area of a
triangle using the 3 side lengths.
Therefore, you do not have to rely on the formula for
area that uses base and height.
You only have to know the perimeter of
the triangle and the side’s measurement.
INTRODUCTION.
4. This formula was founded by a Greek mathematician
called Heron in the year 60 A.D.
It was invented to find the area of triangles.
This was also mentioned in this book named Metrica.
It was also mentioned in the book of the scientist
Archimedes but only in Heron’s book was it properly
understood by people.
That’s why all the credit of the
formula is given to Heron.
INTRODUCTION.
5. Heron of Alexandria (c. 10–70 AD) was an
ancient Greek mathematician and engineer who
was active in his native city of Alexandria, Roman
Egypt.
He is considered the greatest experimenter of
antiquity and his work is representative of the
Hellenistic scientific tradition.
Heron published a well recognized
description of a steam-power
device called an ’’aeolipile” .
WHO WAS HERON ?
6. Heron was also known as the hero of Alexandria or
just hero.
There were more than 18 scientists known as hero at
that time.
Much of Hero's original writings and designs have
been lost, but some of his works were preserved in
Arab manuscripts.
He is said to have been a follower of the Atomists.
Some of his ideas were derived from the works
of Ctesibius.
ABOUT HERON.
7. He had at least 13 works on mathematics,
mechanics and physics.
He devised many mechanical devices which actually
worked practically. One of them was his steam
engine.
He made and wrote a method approximating square
and cube roots of numbers that are not perfect
squares or cubes.
He was also one of the first to make a fountain
which shoots jets of water upwards.
MORE ABOUT HERON.
9. AREA OF SCALENE TRIANGLE
In this triangle it is
impossible to find the
height which is
necessary to find the
are by the formula:
½ x (height) x(base)
Hence we need the
Heron’s formula…..
10. Heron’s formula was actually like this
I. Square root of (s (s-a)(s-b)(s-c) )
II. Where s stood for semi- perimeter.
III. a ,b and c stood for the sides.
IV. First you have to know the complete perimeter of
the triangle.
THE FORMULA
11. Heron’s formula can be used to find out the area of a
triangle in case its height is unknown.
The formula can be used for a scalene triangle in
which the height doesn’t definitely exist. The
formulas can also be used to find the area of
rhombus when only one of the diagonal and the
perimeter are only known .
There are many places where the formula can be
used for example the area of flyovers.
THE USE OF THIS FORMULA.
12. The formula is credited to Heron (or Hero) of
Alexandria, and a proof can be found in his
book, Metrica, written c. A.D. 60. It has been
suggested that Archimedes knew the formula over
two centuries earlier, and since Metrica is a
collection of the mathematical knowledge
available in the ancient world, it is possible that the
formula predates the reference given in that work.
A formula equivalent to Heron's namely:
, where was
discovered by the Chinese independently of the
Greeks. It was published in Shushu
Jiuzhang ,written by Qin Jiushao and published
in A.D. 1247.
13. The heron’s formula has not always been root of (s
(s-a)(s-b)(s-c) )
It has been changed from a simple two step formula
to this current formula.
The first formula that Heron worked through and
founding of the current formula are different.
In the next slide this will be explained .
The first derivation of the formula requires a lot of
higher studies so I will only explain the basic.
EVOLUTION OF THE FORMULA
14. The Heron’s formula has many other methods in
which it can be written in these are a few methods –
A = ¼ square root of (a+b+c)(-a+b+c)(a-b+c)(a+b-c).
A = ¼ square root of 2(a^2*b^2+a^2*c^2+c^2*b^2)-
(a^4+b^4+c^4)
It was derived from the Chinese in the form of –
A=1/2 square root of a^2*c^2-(a^2+c^2-b^2/2)
Since there was no trigonometry at that time it was
very hard to prove the formula back in heron’s time.
THE CHANGING OF THE FORMULA.
15. We can only prove Heron’s formula through advanced
techniques so most of us wouldn’t understand it but we
can also try it with Pythogaros theorem.
By the Pythagorean theorem we have b^2=h^2+d^2 and
a^2=h^2+(c-d)^2 according to the figure at the right.
Subtracting these yields a^2-b^2=c^2-2cd. Thus
d=frac{-a^2+b^2+c^2}{2c}.
Then we get for the height of the triangle that
h^2 & = b^2-d^2=left(frac{2bc}{2c}right)^2-
left(frac{-a^2+b^2+c^2}{2c}right)^2
THE PROOF OF THIS FORMULA.
17. We now apply this result to the formula for the area
A of the triangle that involves a height, in this case
height h from side c:
A & = {ch}{2}
& = sqrt{frac{c^2}{4}cdot frac{4s(s-a)(s-b)(s-
c)}{c^2}}
& = sqrt{s(s-a)(s-b)(s-c)}
CONTINUATION
18. • 1) Find the area of a triangle having sides :
AB = 4 cm
BC = 3 cm
CD = 5 cm
EXAMPLES :
21. 2) Rahul has a garden, which is triangular in shape.
The sides of the garden are 13 m, 14 m, and 15 m
respectively. He wants to spread fertilizer in the
garden and the total cost required for doing it is Rs 10
per m2. He is wondering how much money will be
required to spread the fertilizer in the garden
EXAMPLE 2:
22. • Given a = 13 m , b = 14 m and c = 15 m
So , we will find
the area of the
triangle by
using Heron’s
formula.
SOLUTION OF EXAMPLE 2)
24. Given the rate = Rs 10 per sq. m .
Now :
Total cost = Rs. 10 * 84 = Rs 840/-
CONTINUE …
25. What equilateral triangle would have the same area as a
triangle with sides 6, 8 and 10?
This is not the same as the normal questions and will require
to know the area of the area of the equilateral triangle.
CONCEPT BASED QUESTION
26. HOW TO FIND THE AREA OF AN EQUILATERAL
TRIANGLE
27. First of all we will find the area of the triangle having sides : a
= 6 units , b = 8 units and c = 10 units
SOLUTION