Gives the Area of a Triangle in terms of the lengths of the sides , , and and the Semiperimeter

(1) |

(2) |

(3) | |||

(4) | |||

(5) |

gives the particularly pretty form

(6) |

The proof of this fact was discovered by Heron (ca. 100 BC-100 AD), although it was already known to Archimedes prior to 212 BC (Kline 1972). Heron's proof (Dunham 1990) is ingenious but extremely convoluted, bringing together a sequence of apparently unrelated geometric identities and relying on the properties of Cyclic Quadrilaterals and Right Triangles.

Heron's proof can be found in Proposition 1.8 of his work *Metrica.* This manuscript had been lost for centuries until
a fragment was discovered in 1894 and a complete copy in 1896 (Dunham 1990, p. 118). More recently, writings of the Arab
scholar Abu'l Raihan Muhammed al-Biruni have credited the formula to Heron's predecessor Archimedes (Dunham
1990, p. 127).

A much more accessible algebraic proof proceeds from the Law of Cosines,

(7) |

(8) |

(9) | |||

(10) | |||

(11) | |||

(12) |

(Coxeter 1969). Heron's formula contains the Pythagorean Theorem.

**References**

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, p. 12, 1969.

Dunham, W. ``Heron's Formula for Triangular Area.'' Ch. 5 in
*Journey Through Genius: The Great Theorems of Mathematics.* New York: Wiley, pp. 113-132, 1990.

Kline, M. *Mathematical Thought from Ancient to Modern Times.* New York: Oxford University Press, 1972.

Pappas, T. ``Heron's Theorem.'' *The Joy of Mathematics.* San Carlos, CA: Wide World Publ./Tetra, p. 62, 1989.

© 1996-9

1999-05-25