Trirectangular tetrahedron

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A trirectangular tetrahedron can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin, like:
x>0
y>0
z>0
and x/a+y/b+z/c<1

In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the right angle of the trirectangular tetrahedron and the face opposite it is called the base. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron.

Metric formulas[edit]

If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume

V=\frac{abc}{6}.

The altitude h satisfies[1]

\frac{1}{h^2}=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}.

The area T_0 of the base is given by[2]

T_0=\frac{abc}{2h}.

De Gua's theorem[edit]

Main article: De Gua's theorem

If the area of the base is T_0 and the areas of the three other (right-angled) faces are T_1, T_2 and T_3, then

T_0^2=T_1^2+T_2^2+T_3^2.

This is a generalization of the Pythagorean theorem to a tetrahedron.

See also[edit]

References[edit]

  1. ^ Eves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41.
  2. ^ Gutierrez, Antonio, "Right Triangle Formulas", [1]

External links[edit]