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In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex.The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.==Regular tetrahedron==A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation.The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.===Formulas for a regular tetrahedron===Tetrahedral angle redirects here-->The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin::(±1, 0, −1/√2):(0, ±1, 1/√2)Another set of coordinates are based on an alternated cube with edge length 2. The tetrahedron in this case has edge length 2\sqrt. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.:Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1):Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)For a regular tetrahedron of edge length ''a'':a^2\,|-|Surface areaCoxeter, Harold Scott MacDonald; ''Regular Polytopes'', Methuen and Co., 1948, Table I(i)|align=center|A=4\,A_0=\over3}a=\sqrt\,a\,|-|Edge to opposite edge distance|align=center|l= A_0h =a^3=\,|-|Face-vertex-edge angle|align=center|\arccos\left()\,(approx. 54.7356°)|-|Face-edge-face angle|align=center|\arccos\left(\right) = \arctan(2\sqrt)\,(approx. 70.5288°)|-|Edge central angle,("Angle Between 2 Legs of a Tetrahedron" ), Maze5.net(Valence Angle of the Tetrahedral Carbon Atom ) W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145 known as the ''tetrahedral angle''|align=center|\arccos\left(\right ) = 2\arctan(\sqrt)\,(approx. 109.4712°)|-|Solid angle at a vertex subtended by a face|align=center|\arccos\left(\right)(approx. 0.55129 steradians)|-|Radius of circumsphere|align=center|R=a=\sqrt\,a\,|-|Radius of insphere that is tangent to faces|align=center|r=R==}\,|-|Distance to exsphere center from the opposite vertex|align=center|d_=a=Note that with respect to the base plane the slope of a face (\scriptstyle 2 \sqrt) is twice that of an edge (\scriptstyle \sqrt), corresponding to the fact that the ''horizontal'' distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if ''C'' is the centroid of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex.
The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.〔
For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.
==Regular tetrahedron==
A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.
In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.
Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation.
The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.
===Formulas for a regular tetrahedron===
The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin:
:(±1, 0, −1/√2)
:(0, ±1, 1/√2)
Another set of coordinates are based on an alternated cube with edge length 2. The tetrahedron in this case has edge length . Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.
:Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1)
:Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)
For a regular tetrahedron of edge length ''a'':
a^2\,
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|Surface area〔Coxeter, Harold Scott MacDonald; ''Regular Polytopes'', Methuen and Co., 1948, Table I(i)〕
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|Edge to opposite edge distance
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|Face-vertex-edge angle
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(approx. 54.7356°)
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|Face-edge-face angle〔
|align=center|
(approx. 70.5288°)
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|Edge central angle,〔("Angle Between 2 Legs of a Tetrahedron" ), Maze5.net〕〔(Valence Angle of the Tetrahedral Carbon Atom ) W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145〕 known as the ''tetrahedral angle''
|align=center|
(approx. 109.4712°)
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|Solid angle at a vertex subtended by a face
|align=center|
(approx. 0.55129 steradians)
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|Radius of circumsphere〔
|align=center|
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|Radius of insphere that is tangent to faces〔
|align=center|
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|Distance to exsphere center from the opposite vertex
|align=center|
抄文引用元・出典: フリー百科事典『 regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation.The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.===Formulas for a regular tetrahedron===Tetrahedral angle redirects here-->The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin::(±1, 0, −1/√2):(0, ±1, 1/√2)Another set of coordinates are based on an alternated cube with edge length 2. The tetrahedron in this case has edge length 2\sqrt. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.:Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1):Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)For a regular tetrahedron of edge length ''a'':a^2\,|-|Surface areaCoxeter, Harold Scott MacDonald; ''Regular Polytopes'', Methuen and Co., 1948, Table I(i)|align=center|A=4\,A_0=\over3}a=\sqrt\,a\,|-|Edge to opposite edge distance|align=center|l= A_0h =a^3=\,|-|Face-vertex-edge angle|align=center|\arccos\left()\,(approx. 54.7356°)|-|Face-edge-face angle|align=center|\arccos\left(\right) = \arctan(2\sqrt)\,(approx. 70.5288°)|-|Edge central angle,("Angle Between 2 Legs of a Tetrahedron" ), Maze5.net(Valence Angle of the Tetrahedral Carbon Atom ) W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145 known as the ''tetrahedral angle''|align=center|\arccos\left(\right ) = 2\arctan(\sqrt)\,(approx. 109.4712°)|-|Solid angle at a vertex subtended by a face|align=center|\arccos\left(\right)(approx. 0.55129 steradians)|-|Radius of circumsphere|align=center|R=a=\sqrt\,a\,|-|Radius of insphere that is tangent to faces|align=center|r=R==}\,|-|Distance to exsphere center from the opposite vertex|align=center|d_=a=Note that with respect to the base plane the slope of a face (\scriptstyle 2 \sqrt) is twice that of an edge (\scriptstyle \sqrt), corresponding to the fact that the ''horizontal'' distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if ''C'' is the centroid of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).">ウィキペディア（Wikipedia）』
■regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation.The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.===Formulas for a regular tetrahedron===Tetrahedral angle redirects here-->The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin::(±1, 0, −1/√2):(0, ±1, 1/√2)Another set of coordinates are based on an alternated cube with edge length 2. The tetrahedron in this case has edge length 2\sqrt. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.:Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1):Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)For a regular tetrahedron of edge length ''a'':a^2\,|-|Surface areaCoxeter, Harold Scott MacDonald; ''Regular Polytopes'', Methuen and Co., 1948, Table I(i)|align=center|A=4\,A_0=\over3}a=\sqrt\,a\,|-|Edge to opposite edge distance|align=center|l= A_0h =a^3=\,|-|Face-vertex-edge angle|align=center|\arccos\left()\,(approx. 54.7356°)|-|Face-edge-face angle|align=center|\arccos\left(\right) = \arctan(2\sqrt)\,(approx. 70.5288°)|-|Edge central angle,("Angle Between 2 Legs of a Tetrahedron" ), Maze5.net(Valence Angle of the Tetrahedral Carbon Atom ) W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145 known as the ''tetrahedral angle''|align=center|\arccos\left(\right ) = 2\arctan(\sqrt)\,(approx. 109.4712°)|-|Solid angle at a vertex subtended by a face|align=center|\arccos\left(\right)(approx. 0.55129 steradians)|-|Radius of circumsphere|align=center|R=a=\sqrt\,a\,|-|Radius of insphere that is tangent to faces|align=center|r=R==}\,|-|Distance to exsphere center from the opposite vertex|align=center|d_=a=Note that with respect to the base plane the slope of a face (\scriptstyle 2 \sqrt) is twice that of an edge (\scriptstyle \sqrt), corresponding to the fact that the ''horizontal'' distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if ''C'' is the centroid of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).">ウィキペディアで「In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex.The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.==Regular tetrahedron==A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation.The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.===Formulas for a regular tetrahedron===Tetrahedral angle redirects here-->The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin::(±1, 0, −1/√2):(0, ±1, 1/√2)Another set of coordinates are based on an alternated cube with edge length 2. The tetrahedron in this case has edge length 2\sqrt. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.:Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1):Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)For a regular tetrahedron of edge length ''a'':a^2\,|-|Surface areaCoxeter, Harold Scott MacDonald; ''Regular Polytopes'', Methuen and Co., 1948, Table I(i)|align=center|A=4\,A_0=\over3}a=\sqrt\,a\,|-|Edge to opposite edge distance|align=center|l= A_0h =a^3=\,|-|Face-vertex-edge angle|align=center|\arccos\left()\,(approx. 54.7356°)|-|Face-edge-face angle|align=center|\arccos\left(\right) = \arctan(2\sqrt)\,(approx. 70.5288°)|-|Edge central angle,("Angle Between 2 Legs of a Tetrahedron" ), Maze5.net(Valence Angle of the Tetrahedral Carbon Atom ) W.E. Brittin, J. Chem. Educ., 1945, 22 (3), p 145 known as the ''tetrahedral angle''|align=center|\arccos\left(\right ) = 2\arctan(\sqrt)\,(approx. 109.4712°)|-|Solid angle at a vertex subtended by a face|align=center|\arccos\left(\right)(approx. 0.55129 steradians)|-|Radius of circumsphere|align=center|R=a=\sqrt\,a\,|-|Radius of insphere that is tangent to faces|align=center|r=R==}\,|-|Distance to exsphere center from the opposite vertex|align=center|d_=a=Note that with respect to the base plane the slope of a face (\scriptstyle 2 \sqrt) is twice that of an edge (\scriptstyle \sqrt), corresponding to the fact that the ''horizontal'' distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if ''C'' is the centroid of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).」の詳細全文を読む
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