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