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where s is the semiperimeter of the triangle. It is calculated by
Heron's formula can also be written as
Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin.
Let ΔABC be the triangle with sides a=7, b=4 and c=5. The semiperimeter is , and the area is
The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, Metrica, written c. A.D. 60. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.
A formula equivalent to Heron's, namely
A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows. Let a, b, c be the sides of the triangle and α, β, γ the angles opposite those sides. We have
by the law of cosines. From this proof get the algebraic statement that
The altitude of the triangle on base a has length b·sin γ, and it follows
The difference of two squares factorization was used in two different steps.
By the Pythagorean theorem we have and according to the figure at the right. Subtracting these yields . Thus
Then we get for the height of the triangle that
Taking the square root to get the height and inserting the expression into the area formula yields
from which Heron's formula follows.
but, since the sum of the half-angles is , the triple cotangent identity applies, so the first of these is
Combining the two, we get
from which the result follows.
The brackets in the above formula are required in order to prevent numerical instability in the evaluation.
Three other area formulas have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a, b, and c respectively as ma, mb, and mc and their semi-sum (ma + mb + mc)/2 as σ, we have
Next, denoting the altitudes from sides a, b, and c respectively as ha, hb, and hc, and denoting the semi-sum of the reciprocals of the altitudes as we have
Finally, denoting the semi-sum of the angles' sines as S = [(sin α) + (sin β) + (sin γ)]/2, we have
where D is the diameter of the circumcircle:
Heron's formula is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral. Heron's formula and Brahmagupta's formula are both special cases of Bretschneider's formula for the area of a quadrilateral. Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero.
Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
If U, V, W, u, v, w are lengths of edges of the tetrahedron (first three form a triangle; u opposite to U and so on), then