Acid strength: Difference between revisions
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en>Dirac66 Perhaps a little clearer as requested on talk page. |
en>Dirac66 →Strong acids: 1.CH3SO3H - added reference from CH3SO3H article mentioned in last edit summary 2.FSO3H and triflic acid - added values from same reference |
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[[File:2D-simplex.svg|thumb|A trirectangular tetrahedron can be constructed by a coordinate [[octant (solid geometry)|octant]] and a plane crossing all 3 axes away from the origin, like:<BR>x>0<BR>y>0<BR>z>0<BR>and x/a+y/b+z/c<1]] | |||
In [[geometry]], a '''trirectangular tetrahedron''' is a [[tetrahedron]] where all three face angles at one [[Vertex (geometry)|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== | |||
If the legs have lengths ''a, b, c'', then the trirectangular tetrahedron has the [[volume]] | |||
:<math>V=\frac{abc}{6}.</math> | |||
The altitude ''h'' satisfiy<ref>Eves, Howard Whitley, "Great moments in mathematics (before 1650)", ''Mathematical Association of America'', 1983, p. 41.</ref> | |||
:<math>\frac{1}{h^2}=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}.</math> | |||
The area <math>T_0</math> of the base is given by<ref>Gutierrez, Antonio, "Right Triangle Formulas", [http://gogeometry.com/pythagoras/right_triangle_formulas_facts.htm]</ref> | |||
:<math>T_0=\frac{abc}{2h}.</math> | |||
==De Gua's theorem== | |||
{{main|De Gua's theorem}} | |||
If the [[area]] of the base is <math>T_0</math> and the areas of the three other (right-angled) faces are <math>T_1</math>, <math>T_2</math> and <math>T_3</math>, then | |||
:<math>T_0^2=T_1^2+T_2^2+T_3^2.</math> | |||
This is a generalization of the [[Pythagorean theorem]] to a tetrahedron. | |||
==See also== | |||
*[[Standard simplex]] | |||
*[[Disphenoid]] | |||
*[[Goursat tetrahedron]] | |||
*[[Orthocentric tetrahedron]] | |||
*[[Schläfli orthoscheme]] | |||
== References == | |||
{{reflist}} | |||
== External links == | |||
*{{MathWorld | title = Trirectangular tetrahedron | urlname = TrirectangularTetrahedron}} | |||
[[Category:Polyhedra]] | |||
Revision as of 22:59, 3 February 2014
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
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume
The altitude h satisfiy[1]
The area of the base is given by[2]
De Gua's theorem
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. If the area of the base is and the areas of the three other (right-angled) faces are , and , then
This is a generalization of the Pythagorean theorem to a tetrahedron.
See also
References
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External links
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