Hero's Formula
Hero's formula, also called Heron's formula, relates the area of a triangle to the measures of its three sides. Allowing a, b, and c to denote the lengths of the sides of a triangle, s, the semi-perimeter of the triangle, becomes s = (a + b + c)/2. Hero's formula states simply that the area of a triangle, A, can be expressed as A = sqrt [s(s - a)(s - b)(s - c)]. As a result, the area of the triangle can be determined without knowing its perpendicular height, also known as the triangle's altitude.
Accredited to the Greek geometer Heron (c. 62 AD ), the formula is derived in Heron's most important manuscript, Metrica. Heron's work was discovered in fragmentary form in 1894 and recovered fully in 1896. Recent scholarship, however, suggests that Archimedes (c. 212 BC) may have previously derived the formula.
Although the formula itself is deceptively simple, Heron's original proof is both clever and arduous, utilizing the properties of inscribed quadrilaterals (cyclic quadrilaterals) and right triangles. Today, there are numerous elegant algebraic, geometric, and trigonometric proofs for Hero's formula.
One geometric proof of Hero's formula involves inscribing a circle inside a triangle, and then summing the areas of the three triangles formed by connecting each vertex of the triangle to the center of the circle. Since radii of the inscribed circle are perpendicular to the points of tangency, segment extensions and properties of similar triangles can be used to complete the proof.
Trigonometric and algebraic proofs of Hero's formula generally use the law of cosines, cos C = (a2 + b2 - c2)/2ab. One can then solve for sin C and use the identity that the area of a triangle is equal to (1/2) * a * b * sin C to complete the proof.
Hero's formula is particularly useful when attempting to determine the area of a triangle for which the length of the sides--but not the perpendicular height--is known, making it ideal for use by surveyors and natural scientists collecting data in the field, such as ecologists and wildlife experts. Indeed, Hero noted the appropriateness of this formula for surveying land in his work Dioptra and even developed an instrument called the diopter for the same purpose. Contemporary surveyors use a device very similar-and based on the same mathematical principles--as Hero's diopter.
The side-side-side theorem from Euclidean geometry states that the triangle is uniquely determined by its side lengths--that is, if a, b, and c are the lengths of a triangle's sides, there exists exactly one triangle that can be composed from these segments, and that all other triangles composed of equal segments will be congruent to the original. This uniqueness provides that any information about the triangle--area, inradius (the radius of the circle inscribed by a triangle), circumradius (the radius of the circle inscribing the triangle), etc.--can be determined by knowing only the lengths of its sides. However, the "synthetic" proof of the side-side-side theorem (that is, a proof based on deduction from geometric axioms) does not suit the algebraic needs of a surveyor who wants to actually compute the area. Thus Hero's formula, though it appears more complicated than the side-side-side theorem, fills a practical gap that the other cannot.
Additionally, Hero's formula can be used to solve for the perpendicular height(s) of a triangle given only the measures of its sides through simple algebraic manipulation, which allows one to easily determine the shortest distance from the vertex of a triangular area to its base. Applying Hero's formula to the special case of a right triangle leads to the Pythagorean Theorem of c2 = a2 + b2 (that is, the square of the hypotenuse of a right triangle is equal to the sum of the squares of its sides). The Pythagorean Theorem, then, can be considered a degenerate case of Hero's formula.
This is the complete article, containing 634 words
(approx. 2 pages at 300 words per page).
