In geometryHeron's formula or Hero's formula gives the area of a triangle in terms of the three side lengths abc. It is named after first-century engineer Heron of Alexandria or Hero who Blandade frågor it in his work Metricathough it was probably known centuries earlier.
This triangle's semiperimeter is. In this example, the side lengths and area are integersmaking it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers. Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways.
After expansion, the expression under the square root is a quadratic polynomial of the squared side lengths a 2b 2c 2. The same relation can be expressed using the Cayley—Menger determinant. The formula is credited to Heron or Hero of Alexandria fl. Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier, [3] 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.
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, [6] or as a special case of De Gua's theorem for the particular case of acute triangles[7] or as a special case of Brahmagupta's formula for the case of a degenerate cyclic quadrilateral.
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows. Applying the law of cosines we get. The altitude of the triangle on base a has length b sin γand it follows. The following proof is very similar to one given by Raifaizen.
This equation allows us to härledning herons formel d in terms of the sides of the triangle:. By replacing d with the formula given above and applying the difference of squares identity we get. If r is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude r and bases aband c.
Their combined area is. Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating-point arithmetic. The brackets in the above formula are required in order to prevent numerical instability in the evaluation.
Three other formulae for the area of a general triangle have a similar structure as Heron's formula, expressed in terms of different variables. 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.