Heronian tetrahedron
Encyclopedia
A Heronian tetrahedron is a tetrahedron
whose side lengths, face areas and volume are all rational number
s. The faces must therefore all be Heronian triangle
s.
A regular tetrahedron with rational sides is not a Heronian tetrahedron because its face areas and volume are not rational numbers. A Heronian tetrahedron is sometimes called a perfect tetrahedron.
117 is the smallest possible length of the longest side of a perfect tetrahedron. Its other sidelengths are 51, 52, 53, 80 and 84.
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
whose side lengths, face areas and volume are all rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. The faces must therefore all be Heronian triangle
Heronian triangle
In geometry, a Heronian triangle is a triangle whose sidelengths and area are all rational numbers. It is named after Hero of Alexandria. Any such rational triangle can be scaled up to a corresponding triangle with integer sides and area, and often the term Heronian triangle is used to refer to...
s.
A regular tetrahedron with rational sides is not a Heronian tetrahedron because its face areas and volume are not rational numbers. A Heronian tetrahedron is sometimes called a perfect tetrahedron.
117 is the smallest possible length of the longest side of a perfect tetrahedron. Its other sidelengths are 51, 52, 53, 80 and 84.