### Heron’s Theorem: a Tool for Surveyors

Heron was one of the great Greek mathematicians of Alexandria, following in the tradition of Euclid, Archimedes, Eratosthenes and Apollonius. He lived in the first century, from about AD 10 to AD 70. His interests were in practical rather than theoretical mathematics and he wrote on measurement, mechanics and engineering. He devised a steam-powered device and a wind-wheel that operated an organ. He is regarded as the greatest experimenter of antiquity, but it is for a theorem in pure geometry that mathematicians remember him today. Heron of Alexandria. Triangle of sides a, b and c and altitude h.

Heron’s theorem – or sometimes Hero’s theorem – gives the area of any triangle in terms of the lengths a, b and c of its three sides. The more well-known formula gives the area as half the product of the base c and height h, that is

A = ½ c h

But in many practical contexts the perpendicular height h is not known, whereas the lengths of the sides are. Then Heron’s theorem gives the area of a triangle with sides a, b and c by the formula

[ s ( sa ) ( sb ) ( sc ) ]1/2

where s = ½ ( a + b + c ) is the “semiperimeter”, half the sum of the sides. We do not need to determine the altitude of the triangle.

A Tool for Surveyors

Although the word “geometry” means measurement of the earth, classical geometry was abstract and far removed from practical concerns. Heron’s result was eminently practical and was of direct use in surveying. For example, if we are measuring land area, it is a simple matter to triangulate a polygonal plot. We can easily measure the length of each side. Heron’s formula then gives the area of each triangle and the total area is the sum of the areas of the triangular elements.

The theorem is a proposition in Heron’s Metrica. This work was mentioned by a sixth century scholar Eutocius, but the work itself was lost until 1894 when a fragment was found by a mathematical historian, Paul Tannery, in in a thirteenth century Persian manuscript. Just two years later, another scholar, R. Schöne found a complete manuscript of Metrica in Constantinople, so we are fortunate to have access to this remarkable work today.

Heron’s proof of his result is elementary but quite intricate. It is an ingenious example of abstract geometric reasoning. For details see Dunham (1991, Ch. 5).

Generalizations

Can we extend Heron’s theorem to quadrilaterals and other polygons? Not directly: a triangle is rigid: given the lengths of the sides, the shape – and therefore the area – is uniquely determined. This is obviously untrue for polygons in general: four equal sides a can form a square of area a2, or a rhombus of any area less than a2.

But what about regular polygons? There is a general result that the area of a regular N-gon is equal to half the product of apothem and perimeter length. The apothem is the perpendicular distance from the centre to a side (or the radius r of the inscribed circle). This result is easily shown by adding the areas of N triangles, each having its base on a side and its vertex at the circum-centre (with radius R).

We can also show by elementary trigonometry that the area of the N-gon is

A = π r 2 . [ sin( 2 π / N ) ] / ( 2 π / N )

which, of course, tends to π r 2 as N tends to infinity.

We could also consider how the result generalizes in the case of spherical geometry. That is for another day.

Sources.

Dunham, William, 1991: Journey Through Genius: The Great Theorems of Mathematics. Penguin Books, ISBN: 978-0-14-014739-1.