# 四面體

（點選觀看旋轉模型）

3 | 2 3

Y3

4
6

C3, [3]+, (33)

 四面體（對偶多面體）

## 性质

${\displaystyle {\begin{vmatrix}-1&\cos {(\alpha _{12})}&\cos {(\alpha _{13})}&\cos {(\alpha _{14})}\\\cos {(\alpha _{12})}&-1&\cos {(\alpha _{23})}&\cos {(\alpha _{24})}\\\cos {(\alpha _{13})}&\cos {(\alpha _{23})}&-1&\cos {(\alpha _{34})}\\\cos {(\alpha _{14})}&\cos {(\alpha _{24})}&\cos {(\alpha _{34})}&-1\\\end{vmatrix}}=0\,}$

### 体积

${\displaystyle V={\frac {1}{3}}A_{0}\,h\,}$

${\displaystyle V={\frac {|(\mathbf {a} -\mathbf {d} )\cdot [(\mathbf {b} -\mathbf {d} )\times (\mathbf {c} -\mathbf {d} )]|}{6}}.}$

${\displaystyle V={\frac {|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}{6}},}$

${\displaystyle 6\cdot V={\begin{vmatrix}\mathbf {a} &\mathbf {b} &\mathbf {c} \end{vmatrix}}}$ 或者 ${\displaystyle 6\cdot V={\begin{vmatrix}\mathbf {a} \\\mathbf {b} \\\mathbf {c} \end{vmatrix}}}$ 这里像 ${\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})\,}$ 可以被表示为横或纵向量。

${\displaystyle 36\cdot V^{2}={\begin{vmatrix}\mathbf {a^{2}} &\mathbf {a} \cdot \mathbf {b} &\mathbf {a} \cdot \mathbf {c} \\\mathbf {a} \cdot \mathbf {b} &\mathbf {b^{2}} &\mathbf {b} \cdot \mathbf {c} \\\mathbf {a} \cdot \mathbf {c} &\mathbf {b} \cdot \mathbf {c} &\mathbf {c^{2}} \end{vmatrix}}}$ 这里 ${\displaystyle \mathbf {a} \cdot \mathbf {b} =ab\cos {\gamma }}$ 等。

${\displaystyle V={\frac {abc}{6}}{\sqrt {1+2\cos {\alpha }\cos {\beta }\cos {\gamma }-\cos ^{2}{\alpha }-\cos ^{2}{\beta }-\cos ^{2}{\gamma }}},\,}$

${\displaystyle 288\cdot V^{2}={\begin{vmatrix}0&1&1&1&1\\1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\\1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\\1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2}&0\end{vmatrix}}}$

#### 海伦公式形态的四面体体积公式

${\displaystyle V={\frac {\sqrt {\,(-a+b+c+d)\,(a-b+c+d)\,(a+b-c+d)\,(a+b+c-d)}}{192\,u\,v\,w}}}$

{\displaystyle {\begin{aligned}a&={\sqrt {xYZ}}\\b&={\sqrt {yZX}}\\c&={\sqrt {zXY}}\\d&={\sqrt {xyz}}\\X&=(w-U+v)\,(U+v+w)\\x&=(U-v+w)\,(v-w+U)\\Y&=(u-V+w)\,(V+w+u)\\y&=(V-w+u)\,(w-u+V)\\Z&=(v-W+u)\,(W+u+v)\\z&=(W-u+v)\,(u-v+W).\end{aligned}}}

#### 利用四面体边之间的距离

${\displaystyle V={\frac {d|[\mathbf {a} \times \mathbf {(b-c)} ]|}{6}}.}$

### 关于四面体性质的其它向量公式

${\displaystyle r={\frac {6V}{|\mathbf {b} \times \mathbf {c} |+|\mathbf {c} \times \mathbf {a} |+|\mathbf {a} \times \mathbf {b} |+|(\mathbf {b} \times \mathbf {c} )+(\mathbf {c} \times \mathbf {a} )+(\mathbf {a} \times \mathbf {b} )|}}\,}$

${\displaystyle R={\frac {|\mathbf {a^{2}} (\mathbf {b} \times \mathbf {c} )+\mathbf {b^{2}} (\mathbf {c} \times \mathbf {a} )+\mathbf {c^{2}} (\mathbf {a} \times \mathbf {b} )|}{12V}}\,}$

${\displaystyle r_{T}={\frac {|\mathbf {a^{2}} (\mathbf {b} \times \mathbf {c} )+\mathbf {b^{2}} (\mathbf {c} \times \mathbf {a} )+\mathbf {c^{2}} (\mathbf {a} \times \mathbf {b} )|}{36V}}\,}$

${\displaystyle 6V=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|.\,}$

${\displaystyle \mathbf {G} ={\frac {\mathbf {a} +\mathbf {b} +\mathbf {c} }{4}}.\,}$

${\displaystyle \mathbf {I} ={\frac {|\mathbf {b} \times \mathbf {c} |\,\mathbf {a} +|\mathbf {c} \times \mathbf {a} |\,\mathbf {b} +|\mathbf {a} \times \mathbf {b} |\,\mathbf {c} }{|\mathbf {b} \times \mathbf {c} |+|\mathbf {c} \times \mathbf {a} |+|\mathbf {a} \times \mathbf {b} |+|\mathbf {b} \times \mathbf {c} +\mathbf {c} \times \mathbf {a} +\mathbf {a} \times \mathbf {b} |}}.\,}$

${\displaystyle \mathbf {O} ={\frac {\mathbf {a^{2}} (\mathbf {b} \times \mathbf {c} )+\mathbf {b^{2}} (\mathbf {c} \times \mathbf {a} )+\mathbf {c^{2}} (\mathbf {a} \times \mathbf {b} )}{2\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}.\,}$

${\displaystyle \mathbf {M} ={\frac {\mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )(\mathbf {b} \times \mathbf {c} )+\mathbf {b} \cdot (\mathbf {c} +\mathbf {a} )(\mathbf {c} \times \mathbf {a} )+\mathbf {c} \cdot (\mathbf {a} +\mathbf {b} )(\mathbf {a} \times \mathbf {b} )}{2\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}.\,}$

${\displaystyle \mathbf {G} =\mathbf {M} +{\frac {1}{2}}(\mathbf {O} -\mathbf {M} )\,}$
${\displaystyle \mathbf {T} =\mathbf {M} +{\frac {1}{3}}(\mathbf {O} -\mathbf {M} )\,}$

${\displaystyle \mathbf {a} \cdot \mathbf {O} ={\frac {\mathbf {a^{2}} }{2}}\quad \quad \mathbf {b} \cdot \mathbf {O} ={\frac {\mathbf {b^{2}} }{2}}\quad \quad \mathbf {c} \cdot \mathbf {O} ={\frac {\mathbf {c^{2}} }{2}}\,}$

${\displaystyle \mathbf {a} \cdot \mathbf {M} ={\frac {\mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )}{2}}\quad \quad \mathbf {b} \cdot \mathbf {M} ={\frac {\mathbf {b} \cdot (\mathbf {c} +\mathbf {a} )}{2}}\quad \quad \mathbf {c} \cdot \mathbf {M} ={\frac {\mathbf {c} \cdot (\mathbf {a} +\mathbf {b} )}{2}}.\,}$

Td
T
[3,3]
[3,3]+
*332
332
24
12

C3v
C3
[3]
[3]+
*33
33
6
3

D2d
S4
[2+,4]
[2+,4+]
2*2
8
4

D2 [2,2]+ 222 4

C2v
=D1h
[2] *22 4

Cs
=C1h
=C1v
[ ] * 2

C2
=D1
[2]+ 22 2

C1 [ ]+ 1 1

### 四面体正弦定理和所有形状四面体所构成的空间

${\displaystyle \sin \angle OAB\cdot \sin \angle OBC\cdot \sin \angle OCA=\sin \angle OAC\cdot \sin \angle OCB\cdot \sin \angle OBA.\,}$

## 其他種類的四面體

### 三角錐

#### 正三角錐

 錐高較低的正三角錐 正四面體 錐高較高的正三角錐 斜正三角錐

### 四面形

類別 （點選觀看旋轉模型） 多面形、均勻多面體、球面鑲嵌 {2,4} 4 | 2 2 4 4 2 F=4, E=4, V=2 （χ=2） 二角形 24 D4h, [2,4], (*224), order 16 D4, [2,4]+, (224), order 16

### 四面體列表

4 6 4 2 4個正三角形 , A3, [3,3], (*332)

3個三角形側面
C3v, [3], (*33)

t{2,2}
{2}x{}
4 6 4 2 2個二角形
2個矩形
D2h, [2,2], (*222), order 8

{ }+{2}
4 6 4 2 4個三角形 D2h, [2,2], (*222) order 8

h0,1{2,2}
s{2,1}
{ }⨂{1}
2 4 4 2 2個一角形
2個三角形
D1d, [2+,2], (2*1), order 4

{2,4}
2 4 4 2 4個二角形 D4h, [2,4], (*224), order 16

{3,4}/2
{3,4}3
3 6 4 1 4個正三角形 S4, order 24
{4,4}2,1 環形多面體 {4,4}2,1 4 8 4 0 4個正方形
{6,3}2,1 環形多面體 {6,3}2,1 8 8 4 4 4個正六邊形

C1v, [1]

C2v, [2]

C3v, [3]

C4v, [4]

C5v, [5]

C6v, [6]

C7v, [7]

C8v, [8]

C9v, [9]

C10v, [10]
...

C∞v, [∞]

Ciπ/λv, [iπ/λ]

## 註釋

1. 四面體. 教育部重編國語辭典. [2022-12-28]. （原始内容存档于2022-12-28）.
2. 三角錐. 教育部重編國語辭典. [2022-12-28]. （原始内容存档于2022-12-28）.
3. ^ Ace, Tom, The Szilassi polyhedron, [2013-03-03], （原始内容存档于2016-09-07）
4. ^ Arseneva, Elena and Kleist, Linda and Klemz, Boris and Löffler, Maarten and Schulz, André and Vogtenhuber, Birgit and Wolff, Alexander. Adjacency Graphs of Polyhedral Surfaces. arXiv preprint arXiv:2103.09803. 2021.
5. ^ "Simplex Volumes and the Cayley-Menger Determinant"页面存档备份，存于互联网档案馆）, MathPages.com
6. ^ Kahan, William M.; "What has the Volume of a Tetrahedron to do with Computer Programming Languages?"页面存档备份，存于互联网档案馆）, pp. 16–17
7. ^ Simonov, VI and Belov, NV. Characteristics of the crystal structure of rinkite. Soviet Physics Crystallography. 1968, 12 (5): 740–744.
8. ^ The 4-hosohedron. weddslist.com. [2022-12-15]. （原始内容存档于2022-12-17）.