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A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]
The same area formula can also be derived from Heron's formula for the area of a triangle from its three sides. However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between the semiperimeter and side length in those triangles. [19]
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. [ 1 ] [ 2 ] Heronian triangles are named after Heron of Alexandria , based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84 .
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is , which is larger than three times , the area of a 26–14–14 isosceles triangle (all by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.
A Heronian triangle, also known as a Heron triangle or a Hero triangle, is a triangle with integer sides and integer area. All Heronian triangles can be placed on a lattice with each vertex at a lattice point. [7] Furthermore, if an integer triangle can be place on a lattice with each vertex at a lattice point it must be Heronian.
Heron was the flag bearer for Team Panama at the Olympics opening ceremony. She is one of eight Panamanian athletes, and the only gymnast from the country, to participate in the 2024 Paris Olympics.
Heron's formula. Note that the area of triangle ABC is also divided into 6 smaller triangles, also in 3 pairs, with the triangles in each pair having the same area. For example, the two triangles near vertex A, being right triangles of width s − a and height r, each have an area of 1 / 2 r(s − a).