Heron/Hero's Formula

Where:
s = half of the triangle's perimeter (its sides a, b, and c added together and divided by 2):

a = side 1 of the triangle
b = side 2 of the triangle
c = side 3 of the triangle

Hero of Alexandria discovered that by taking the square root of half a triangle's perimeter (referred to as the semiperimeter) multiplied by the difference between the semiperimeter and each separate side, you can find the area of the triangle.

This formula, then, can only be used when all of the sides of the triangle are known.

Where Did the Formula Come From?
It may be possible that Heron developed this formula with the Pythagorean Theorem in mind as many people can prove Heron's Formula with the Pythagorean Theorem and the Pythagorean Theorem with Heron's Formula. This very well could be a coincidence, though.

Visit the following sites for more information on proofs of Heron's Formula:
http://www.cut-the-knot.org/pythagoras/herons.shtml
http://mathworld.wolfram.com/PythagoreanTheorem.html (and scroll down to (12) on the right hand side of the page)

How Can This Formula Be Used Outside of Math Class?
"Who really cares about triangles and how big they are in real life?" you're asking. Well, I have a few answers for you.

Say a building company is plotting the land on which it is going to build a playground. The swings of the playground will be in a separate, triangular area away from the other playground equipment. The school is requesting that mulch be used only in this area. To know how much mulch to use to cover the triangular area, the builders need to know the area of the triangle.

How about a painter who needs to know how much paint to use to cover a side of a staircase? If he/she knows the area of the triangle that the stairs and corner of the wall create, he won't buy too much or too little paint.

Someone writing a report on the Bermuda Triangle or the Polynesian Triangle might want to include in his/her paper how many miles these geographical areas cover.

If someone drew two triangles and used optical illusion to make them look like the same size, a curious witness may want to ease his/her mind by calculating which one is actually larger.

I'm sure there are plenty more instances; this just shows you that the equation for the area of a triangle isn't just another "pointless math equation" that you don't really need to know.

Who is Heron/Hero? (and Which Name is Right?)
Inventor and mathematician of 62 AD, Hero of Alexandria seems to have been a genius of a man. He was one of the only people in his time who was not scared to embrace technology; he realized the value of saving time and effort by letting a machine do certain tasks for men.

The way John Lienhard from the University of Houston's College of Engineering puts it, "Hero's story has a moral: we have no freedom without letting go of control." The men of his time were busy enjoying their control and having authority over others while he was busy becoming famous and one of the greatest inventors of all time.

Hero invented many things that are very well heard of in today's world such as the steam engine, the odometer, a simple fountain, the vending machine, complex string puppets, and many groundbreaking laws of mechanics, science, physics, and mathematics.

His studies were published in five works entitled Pneumatica, Automata, Mechanics, Metrics, and Dioptra. Three of these were originally written in Greek and two in Arabic.

As for what his true name is, no one is really sure. Many refer to him as Hero and others could swear it's Heron.

Example
The lengths of the sides of a triangle are 3, 5, and 8. Using Heron's Formula, find the triangle's area.

First, let's find "s" (our triangle's semiperimeter.) Plug in the values for a, b, and c into our equation:

Do the addition on the top of the equation:
3 + 5 + 4 = 12

That gives us: 12 / 2. Divide 12 / 2 = 6

s = 6

Now, plug the s, a, b, and c values into the master equation:

Solve everything inside the parenthesis that starts with s - ...:
6 - 3 = 3
6 - 5 = 1
6 - 4 = 2

Multiply those three new values together:
(3) (1) (2) = 6

Do the final multiplication:
(6) (6) = 36

Finally we get: sqrt(36). Find the square root: sqrt(36) = 6

The area of our triangle is 6.

Works Cited
Lienhard, John. "Hero of Alexandria." Engines of Our Ingenuity. 1997. The Engines of
Our Ingenuity. 16 June 2006. <http://www.uh.edu/engines/epi1038.htm>.

"Hero of Alexandria." Technology Museum of Thessaloniki. 2001. Technology
Museum of Thessaloniki. 16 June 2006. <http://www.tmth.edu.gr/en/aet/5/55.html>.

"Hero of Alexandria." Wikipedia.com. 14 June 2006. Wikimedia Foundation, Inc. 16
June 2006. <http://en.wikipedia.org/wiki/Heron_of_Alexandria>.

"Heron's Formula." Wikipedia.com. 16 May 2006. Wikimedia Foundation, Inc. 16
June 2006. <http://en.wikipedia.org/wiki/Heron's_formula>.

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