I. A brief history of **Heron**
of Alexandria

II. **Heron's Formula**, including
a GSP sketch to test

III. Three **proofs** of Heron's
Formula: one algebraic, one geometric, and one trigonometric

IV. Related topics:

A.

Brahmagupta's Generalization, including a GSP sketch to test and a proofB. An

extensionof Brahmagupta's Generalization, including a GSP sketch to testC. The

Pythagorean Theorem, including a proof using Heron's Formula

V. **Resources**

Not much is known about the man named Heron of Alexandria. Even his name is not definite; he has been called Heron and Hero. No one knows exactly when he lived, though it is commonly believed that he lived sometime between 150 B.C. and 250 A.D. Heron did live in the great scholarly city of Alexandria, Egypt, where many Greek mathematicians and scientists studied. Yet it is not known whether he was a Greek or actually an Egyptian with Greek training. What is sure, though, is that Heron of Alexandria was a brilliant man who gave the modern world much insight into the mathematical and physical sciences.

Heron wrote so many works on mathematical and physical subjects that "it is customary to described him as an encyclopedic writer in these fields" (Eves, p. 178). Most of these works can be divided into two categories: geometric and mechanical. While approximately fourteen of his treatises have been uncovered, there are references to other lost works.

One of Heron's treatises, called __Pneumatica__, describes
almost one hundred machines and toys, including a fire engine,
a wind organ, and a device for opening temple doors by a fire
on the altar. His __Dioptra__ consists of engineering applications
of an ancient type of surveyor's transit. The work __Catoptrica__
deals with properties and constructions of mirrors.

Perhaps Heron's greatest contribution to the mathematical world
was his work called __Metrica__, which was written in three
books. It was mainly of geometric nature, dealing with area and
volume mensuration of various polygons and solids. It investigated
properties of regular polygons, circles, and conic sections. In
this work Heron also gave a method of finding the approximation
of the square root of a non-square integer; this method is used
by many computers today.

Finally, __Metrica__ contains Heron's proof of the formula
used to find the area of a triangle given the lengths of the three
sides. Most scholars believe the proposition should actually be
attributed to Archimedes. But the formula was given Heron's name,
and it is by the term Heron's Formula that the proposition is
now widely known.

Most schoolchildren know the traditional formula used to find the area of a triangle: A = (1/ 2)*b*h. Obviously, to find the area of a triangle using this formula, one must know the length of a side of the triangle (the base, b) and the length of the altitude to that side (the height, h).

On the other hand, Heron's Formula can be used to find the area of a triangle when one knows the lengths of the three sides. Note that it is not necessary to know a height in order to use this formula.

**Heron's Formula states:**

Given the lengths a, b, and c of the three sides of a triangle...

...and after finding the semiperimeter, s, of the triangle,...

...the area of the triangle can be found using this formula:

Click **here** for
a Geometer's SketchPad file to manipulate and to relate the traditional
area formula to Heron's Formula.

There are many proofs of Heron's Formula. Most can be categorized as algebraic, geometric, or trigonometric. The following list includes a presentation of one proof of each of these types.

Click **here**
to see an algebraic proof of Heron's Formula.

Click **here**
to see a geometric proof of Heron's Formula. This proof is based
on Heron's proof of the formula in __Metrica__.

Click **here**
to see a trigonometric proof of Heron's Formula.

Brahmagupta was a Hindu mathematician who lived in India during
the seventh century A.D. He wrote __Brahma-sphuta-sidd'hanta__,
which was mainly a work on astronomy, but two of its chapters
dealt with mathematics. Included in this work was a formula that
can be used to find the area of a cyclic quadrilateral when given
the lengths of the four sides. The formula is often called Brahmagupta's
Generalization, as opposed to Brahmagupta's Formula, because later
commentators assumed it was a formula to be used to find the area
of *any* quadrilateral. Because they failed to see the limitation
of the formula, the critics found that it did not work in all
cases.

Given a quadrilateral inscribed in a circle, with sides of length a, b, c, and d,...

...and after finding the semiperimeter, s, of the quadrilateral,...

...the area of the quadrilateral can be found using this formula:

***Note that when the distance d equals 0, the cyclic quadrilateral becomes a triangle. Brahmagupta's Generalization then reduces to Heron's Formula.

Click **here** for
a Geometer's SketchPad file to test Brahmagupta's Generalization.

Click **here**
to see a proof of Brahmagupta's Generalization.

Because Brahmagupta's Generalization works only for cyclic quadrilaterals, it is interesting to note that an extension of his formula can be used to find the area of any quadrilateral.

Given a quadrilateral with sides of length a, b, c, and d,...

...let the measure of the angle between sides a and d be A, and the measure of the angle between sides b and c be B.

After finding the semiperimeter, s, of the quadrilateral,...

...the area of the quadrilateral can be found using this formula:

***Note that when the distance d equals 0, the quadrilateral becomes a triangle. This extension of Brahmagupta's Generalization then reduces to Heron's Formula.

Click **here** for
a Geometer's SketchPad file to test this extension of Brahmagupta's
Generalization.

There are many, many ways to prove the Pythagorean Theorem. One way is to use Heron's Formula.

Given a right triangle with legs of length a and b and hypotenuse of length c,...

...the following relationship can be stated:

Click **here** to
see a proof of the Pythagorean Theorem using Heron's Formula.

Coxeter, H.S.M. and S.L. Greitzer. __Geometry Revisited__.
Random House, Inc., New York City, 1967.

Eves, Howard. __An Introduction to the History of Mathematics__,
6th ed. Saunders College Publishing, Orlando, 1990.

Kevin Brown's MathPages: http://www.seanet.com/~ksbrown/kmath196.htm

Cut-the-Knot: http://www.cut-the-knot.com/pythagoras/herons.html

Dr. Jim Wilson's problem solving course: http://jwilson.coe.uga.edu/emt725/Heron/Trig.Heron.html