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

He 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:

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 ):

**Substitute: ** **( 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:

**x**

^{2}+ y^{2}= 1 = Cos^{2}A + Sin^{2}A

Next, to find the distance * a*, between points B and C, use the distance formula, also derived from Pythagoras:

**a**

^{2}= ( x_{2}– x_{1})^{2}+ ( y_{2}– y_{1})^{2}And substitute coordinates **B** ( c Cos A, c Sin A ) and **C **( b, 0 ) for ( x_{2}, y_{2} ) and ( x_{1}, y_{1} ):

**a ^{2 } = **

*( c CosA – b )*

^{2}+ ( c SinA – 0 )^{2}= c^{2}Cos^{2}A – 2bc CosA + b^{2}+ c^{2}Sin^{2}A**=** *c ^{2} Cos^{2}A + c^{2} Sin^{2}A + b^{2} – 2bc CosA*

**= ** *b ^{2} + c^{2} ( Cos^{2}A + Sin^{2}A ) – 2bc CosA *

**a ^{2} = b^{2} + c^{2} – 2bc CosA (This is the Law of Cosines)**

Get Cos A by itself and substitute into ** Cos ^{2} A + Sin^{2} A = 1**

** **therefore, we have written **Sin ^{2} A** in terms of the triangle’s 3 sides,

*,*

**a***, and*

**b***:*

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

**Sin ^{2} A** can now be simplified to include this new term,

*s*, and rewritten as

**Sin A**:

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

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