Ἥρων ὁ Ἀλεξανδρεύς

penrose triangleHero of Alexandria, sometimes known as Heron, lived in Egypt circa 10 – 70 CE, during the Ptolemaic period.

heron-alexandrieHe was a Greek engineer whose inventions included a steam-powered engine, called an “aeolipile,” and devices to harness wind power.

Today, he is best known as the mathematician who formulated a method to calculate the area of any triangle, given only the lengths of three sides, called Hero’s Formula.

Here is how the formula is derived:

First, we know that the Area of a Rectangle = (Length)(Width)

i.e., area is found by knowing 2 dimensions, perpendicular to each other; and

Area of a Triangle = ½ (Base) (Height)

i.e., area is found by knowing 2 dimensions, perpendicular to each other.

An alternative formula for the area of a triangle can be found if we know one of its angles:

heroscanHero began by using a coordinate geometry method by placing a triangle in the coordinate plane and each of its vertices correspond to units along those axes.

Let vertex A be located at ( 0, 0 ), B is located at some distance ( x, y ), and C = ( b, 0 ):

heroscan2Substitute:   ( c Cos A, c Sin A ) for ( x, y )

Because we have an inscribed right triangle, and letting hypotenuse length, c = 1, then, by Pythagorean Theorem:   x2 + y2 =  1  =  Cos2 A + Sin2 A

Next, to find the distance a, between points B and C, use the distance formula, also derived from Pythagoras:   a2  =  ( x2 – x1 )2 + ( y2 – y1 )2

And substitute coordinates B ( c Cos A, c Sin A ) and C ( b, 0 ) for ( x2, y2 ) and ( x1, y1 ):

a2  =   ( c CosA – b )2 + ( c SinA – 0 )2   =   c2 Cos2A – 2bc CosA + b2 + c2 Sin2A

=   c2 Cos2A + c2 Sin2A + b2 – 2bc CosA

 b2 + c2 ( Cos2A + Sin2A ) – 2bc CosA

a2  =   b2 + c2 – 2bc CosA            (This is the Law of Cosines)

Get Cos A by itself and substitute into  Cos2 A + Sin2 A = 1

  heroscan3therefore, we have written Sin2 A in terms of the triangle’s 3 sides, a, b, and c:

heroscan4next, the brilliant substitution for s, which is equal to one-half of the triangle’s perimeter:

heroscan5Sin2 A can now be simplified to include this new term, s, and rewritten as Sin A:

heroscan6Since Area = ½ b c Sin A, substitute Sin A and simplify:

heroscan7This formula for the area of any triangle, given 3 sides, is Hero’s Formula

                                                                                                                                       

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