Difference between revisions of "Tetrahedron"
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Any four points chosen in space will be the vertices of a tetrahedron as long as they do not all lie on a single [[plane]]. | Any four points chosen in space will be the vertices of a tetrahedron as long as they do not all lie on a single [[plane]]. | ||
| − | Regular tetrahedra, in which all edges have equal [[length]] and all faces are [[congruent]] [[equilateral triangle]]s, are one of the five types of [[Platonic solid]]s. | + | <asy> |
| + | import three; | ||
| + | currentprojection = orthographic(-1.2,-0.2,0.4); | ||
| + | triple[] P = {(0,0,(2/3)^.5),(3^(-0.5),0,0),(-1/2/3^.5,1/2,0),(-1/2/3^.5,-1/2,0)}; | ||
| + | void drawFrontFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linewidth(0.7)); | ||
| + | /* fill(P[x] -- P[y] -- P[z] -- cycle, rgb(0.7,0.7,0.7)); */ | ||
| + | } | ||
| + | void drawBackFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linetype("2 6")); | ||
| + | } | ||
| + | drawFrontFace(0,3,2);drawBackFace(0,1,3);drawBackFace(0,2,3);drawBackFace(1,2,3); | ||
| + | </asy> | ||
| + | |||
| + | The volume of a tetrahedron is <math>\frac{1}{3}bh</math>. | ||
| + | |||
| + | Regular tetrahedra, in which all edges have equal [[length]] and all faces are [[congruent]] [[equilateral triangle]]s, are one of the five types of [[Platonic solid]]s. The volume of a regular tetrahedron can also be found via <math>\frac{a^3}{6\sqrt2}</math>, where <math>a</math> is the side length. | ||
The [[polyhedral dual]] of a tetrahedron is another tetrahedron. | The [[polyhedral dual]] of a tetrahedron is another tetrahedron. | ||
Latest revision as of 11:54, 20 February 2024
The tetrahedron (plural tetrahedra) or triangular pyramid is the simplest polyhedron. Tetrahedra have four vertices, four triangular faces and six edges. Three faces and three edges meet at each vertex.
Any four points chosen in space will be the vertices of a tetrahedron as long as they do not all lie on a single plane.
![[asy] import three; currentprojection = orthographic(-1.2,-0.2,0.4); triple[] P = {(0,0,(2/3)^.5),(3^(-0.5),0,0),(-1/2/3^.5,1/2,0),(-1/2/3^.5,-1/2,0)}; void drawFrontFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linewidth(0.7)); /* fill(P[x] -- P[y] -- P[z] -- cycle, rgb(0.7,0.7,0.7)); */ } void drawBackFace(int x, int y, int z){ draw(P[x] -- P[y] -- P[z] -- cycle, linetype("2 6")); } drawFrontFace(0,3,2);drawBackFace(0,1,3);drawBackFace(0,2,3);drawBackFace(1,2,3); [/asy]](http://latex.artofproblemsolving.com/5/8/a/58a906ea42c6080e76d6bfbd11d822a0315523cd.png)
The volume of a tetrahedron is 
.
Regular tetrahedra, in which all edges have equal length and all faces are congruent equilateral triangles, are one of the five types of Platonic solids. The volume of a regular tetrahedron can also be found via 
, where 
is the side length.
The polyhedral dual of a tetrahedron is another tetrahedron.
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