# 四角柱

类别 四种四角柱 柱体 双四角锥 .mw-parser-output .CDD-invert img{filter:invert(100%)}.mw-parser-output .CDD-white img{filter:contrast(0)brightness(100)}.mw-parser-output .CDD-red img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(1000)}.mw-parser-output .CDD-yellow img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(15)hue-rotate(30deg)saturate(10)}.mw-parser-output .CDD-orange img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(10)}.mw-parser-output .CDD-green img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(1000)hue-rotate(120deg)}.mw-parser-output .CDD-cyan img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(15)hue-rotate(180deg)saturate(10)}.mw-parser-output .CDD-blue img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(1000)hue-rotate(-120deg)}.mw-parser-output .CDD-purple img{filter:brightness(0%)contrast(0%)sepia(100%)saturate(15)hue-rotate(-140deg)saturate(10)} t{2,4} or {4}x{} 2 4 | 2 6 12 8 F=6, E=12, V=8 （χ=2） 4个矩形侧面 2个四边形底面 4.4.4 凸

 只要底面是四边形皆称为四角柱

## 性质

### 体积与表面积

V凸四角柱 = ${\displaystyle {\frac {1}{2}}\left|{\begin{vmatrix}x_{{}_{AC}}&y_{{}_{AC}}&z_{{}_{AC}}\\x_{{}_{BD}}&y_{{}_{BD}}&z_{{}_{BD}}\\x_{{}_{PQ}}&y_{{}_{PQ}}&z_{{}_{PQ}}\end{vmatrix}}\right|}$

V凸四角柱 = ${\displaystyle {\frac {1}{2}}\left|{\overrightarrow {PQ}}\cdot ({\overrightarrow {AC}}\times {\overrightarrow {BD}})\right|}$

A底面积${\displaystyle ={\tfrac {1}{2}}{\overline {AC}}\,{\overline {BD}}\sin(\angle APB),}$

${\displaystyle {\overrightarrow {AC}}\times {\overrightarrow {BD}}=\left\|{\overrightarrow {AC}}\right\|\left\|{\overrightarrow {BD}}\right\|\sin(\angle APB)\ \mathbf {n} }$

A凸四角柱 = ${\displaystyle \left|{\overrightarrow {AC}}\times {\overrightarrow {BD}}\right|+(\left|{\overline {AB}}\right|+\left|{\overline {BC}}\right|+\left|{\overline {CD}}\right|+\left|{\overline {DA}}\right|)\cdot \mathbf {h} }$

### 作为截角四面形

 四面形 截角四面形

## 常见的四角柱

 正四角柱

Oh
(*432)
D4h
(*422)
D2h
(*222)

(111)

(112)

(123)

### 长方体

 四角柱 矩形柱的展开图

### 梯形柱

 梯形柱 梯形柱 梯形柱展开图

### 非凸四角柱

 二复合二角形柱体两个四边形二面体组成的复合体在施莱夫利符号中计为{4/2}x{}但是其已退化，不具有体积 凹鹞形柱底面为凹鹞形的柱体

### 斜四角柱

 斜平行四边形柱体又称平行六面体

## 相关多面体与镶嵌

[4,2], (*422) [4,2]+, (422) [1+,4,2], (222) [4,2+], (2*2)
{4,2} t{4,2} r{4,2} 2t{4,2}=t{2,4} 2r{4,2}={2,4} rr{4,2} tr{4,2} sr{4,2} h{4,2} s{2,4}

V42 V82 V42 V4.4.4 V24 V4.4.4 V4.4.8 V3.3.3.4 V22 V3.3.2.3

3 4 5 6 7 8 9 10 11英语Hendecagonal prism 12
[2n,2]
[n,2]
[2n,2+]

t{2,1}

t{2,2}

t{3,2}

{4,2}

t{5,2}

t{6,2}

t{7,2}

t{8,2}
...

t{2,∞}

t{2,iπ/λ}

## 参考文献

1. ^ 杨波. 四棱柱側棱上四點共面的一個充要條件. MIDDLE SCHOOL MATHEMATICS (陕西省城固师范学校). 2003, 10. doi:10.3969/j.issn.1002-7572.2003.10.024.
2. ^ 李汶忠. 四棱柱体积的解析求法. 中央民族大学学报: 自然科学版. 1997, (1): 39-41.
3. ^ 李汶忠. 四边形面积和四棱柱, 锥体积的解析求法. 数学通报. 1985, 6: 12.
4. ^ Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.
5. ^ Josefsson, Martin, Five Proofs of an Area Characterization of Rectangles (PDF), Forum Geometricorum, 2013, 13: 17–21 [2016-08-24], （原始内容 (PDF)存档于2016-03-04）.
6. ^ Wilson, Edwin Bidwell. Vector Analysis: A text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs. Yale University Press. 1901. p. 60–61
7. ^ Dennis G. Zill; Michael R. Cullen. Definition 7.4: Cross product of two vectors. Advanced engineering mathematics 3rd. Jones & Bartlett Learning. 2006: 324. ISBN 0-7637-4591-X.
8. ^ 埃里克·韦斯坦因. Cube. MathWorld.
9. ^ Robertson, Stewart Alexander, Polytopes and Symmetry, Cambridge University Press: 75, 1984, ISBN 978-0-521-27739-6
10. ^ nets of a cuboid. donsteward. 2013-05-24 [2016-08-23]. （原始内容存档于2016-03-04）.
11. ^ TR Smith. Volume of a Trapezoidal Prism: Formula and Examples. Owlcation. 2016-02-08 [2016-08-23]. （原始内容存档于2016-08-26）.
12. ^ どちらも四角柱です. morinogakko. [2016-08-24]. （原始内容存档于2016-08-27）.