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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 ...
A Heronian tetrahedron [1] (also called a Heron tetrahedron [2] or perfect pyramid [3]) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles (named for Hero of Alexandria ).
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
Today, however, his name is most closely associated with Heron's formula for the area of a triangle in terms of its side lengths. Hero also reported on a method for calculating cube roots. [22] In solid geometry, the Heronian mean may be used in finding the volume of a frustum of a pyramid or cone.
The final line can be rewritten to obtain Heron's formula for the area of a triangle given three sides, which was known to Archimedes prior. [ 8 ] In the case of n = 3 {\displaystyle n=3} , the quantity v 3 {\displaystyle v_{3}} gives the volume of a tetrahedron , which we will denote by V {\displaystyle V} .
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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 .
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