四邊形

四邊形
面積不同的四邊形有不同的算法
見下文
內角90 (正方形長方形時)

幾何學中,四邊形是指有四條和四個頂點多邊形,其內角和為360。四邊形有很多種,其中對稱性最高的是正方形,其次是長方形菱形,較低對稱性的四邊形如等腰梯形鷂形,對稱軸只有一條。其他的四邊形依照其類角的性質可以分成凸四邊形和非凸四邊形,其中凸四邊形代表所有內角角度皆小於180度。非凸四邊形可以再進一步分成凹四邊形和複雜四邊形,其中複雜四邊形表示邊自我相交的四邊形。

簡單四邊形

四邊形可以分成簡單四邊形和複雜四邊形兩大類,簡單四邊形表示邊沒有交錯的四邊形,複雜四邊形表示邊有交錯的四邊形。

凸四邊形

凸四邊形是指所有角都比平角小的四邊形,且兩條對角線都落在其內部。

  • 不規則凸四邊形:是凸四邊形中最大的子集,包含了所有的凸四邊形,一般會用任意凸四邊形稱呼之。
  • 不平行四邊形:沒有任何邊互相平行的四邊形。這個四邊形的名稱在英式英文與美式英文中有不同的稱呼,英式英文將之稱為「irregular quadrilateral」,而北美英文英语North American English則稱為「trapezium」。
  • 梯形:只有一雙對邊平行的四邊形。這個四邊形的名稱在英式英文與美式英文中有不同的稱呼,英式英文將之稱為Trapezium,而北美英文英语North American English則稱為trapezoid。
  • 等腰梯形:一雙對邊平行、另外兩邊等長但不平行,也称为圓內接梯形,是有一對平行邊的圓內接四邊形,一種擁有更高的對稱性的梯形。
  • 三等邊梯形:一雙對邊平行、另外兩邊和一底邊等長的梯形。
  • 平行四邊形:具有兩對平行邊的四邊形或兩對邊平行的四邊形。其等效條件是有兩對邊等長、兩對角等角,或者是對角線彼此平分。正方形、長方形、斜方形和菱形都是平行四邊形。
  • 菱形:主流文獻上有兩種定義。較粗疏的定義是四邊相等,在這定義下,正方形是菱形的一種。另外一種定義較嚴謹,菱形是四邊相等,但角不是直角[1]。在這定義下的正方形就不是菱形的一種。
  • 斜方形:對角相等且對邊相等,但邊不全相等且角不是直角的四邊形[1]。換句話說,就是平行四邊形中不是菱形的形狀[2]。其英語名稱為Rhomboid[3],容易與菱形(英語:Rhombus[4]混淆。
  • 矩形:四個角都是直角的四邊形。其等效條件是對角線互相平分且等長。正方形和長方形是矩形的一種。
  • 長方形:角是直角,但四邊不全相等的四邊形[1]
  • 正方形:四邊相等且四個角是直角的四邊形[1]。由於其四個角都等角,又凸四邊形內角和為360度,因此其四個角都是直角。其等效條件是對邊平行且等長,對角線互相垂直平分且等長。
  • 鷂形,相鄰邊等長的四邊形。其中一條對角線可以將之分割成兩個全等的三角形,因此在這對角線兩側的對角會相等,這也意味著其對角線垂直。鷂形又稱鳶形或箏形。

  • 圓內接四邊形:含有外接圓的四邊形,換句話說,這個四邊形的四個頂點落在一個圓上。
  • 圓外切四邊形:含有內切圓的四邊形,換句話說,這個四邊形的四條邊與一個圓相切。
  • 圆外切梯形:有一對平行邊的圓外切四邊形。
  • 雙心四邊形:内切圓在兩對對邊的切點的連線相互垂直,含有外接圓和內切圓。這個四邊形的頂點落在一個圓上且對角和為180度。
  • 直角筝形:有一對直角的鷂形。正鷂形是一種雙心四邊形。
  • 正交四边形:兩對角線垂直的四邊形。
  • 等對角線四邊形:對角線等長的四邊形[5]
  • 旁心四邊形:四條邊向外延伸後能與一個圓心在四邊形外的圓相切的四邊形[6][7]
  • Equilic四邊形:表示有一對邊長度相等,且兩者成60度角的四邊形。
  • 瓦特四邊形:一個對邊等長的四邊形[8]
  • 二次四邊形:是指四個頂點都落在正方形周界上的四邊形[9]
  • 直徑四邊形 :是指有一條邊是外接圓圓心的圆内接四边形[10]

非凸四邊形

簡單四邊形中的非凸四邊形是指不是凸四邊形的其他四邊形。

  • 凹四邊形:是指有至少一個角大於180度的四邊形。
  • 鏢形(或箭頭形凹鷂形):相鄰邊等長的凹四邊形。

複雜四邊形

反平行四邊形是複雜四邊形的一個例子

邊自我相交的四邊形稱為複雜四邊形、折四邊形、交叉四邊形、蝴蝶四邊形或領結四邊形。交叉四邊形在兩個相交邊的四個內角(兩個銳角和兩個優角)內角和可達720度[11]

  • 星形四邊形(或四角星):指邊自相交的一種四邊形,但只能是退化的多邊形,即兩個二角形的複合圖形。
  • 折四邊形:兩對邊相交的四邊形。
  • 反平行四邊形:兩對邊等長的折四邊形。
  • 交叉矩形:有一對邊平行且其對角線和平行的對邊可以形成一個矩形的反平行四邊形
  • 交叉正方形:有一對邊平行且交叉的對邊互相垂直[12]

分類

分類依據 根據對稱的特性 根據四邊長度: 根據角度大小: 根據邊的情形: 根據頂點的情形:
種類
  • 一條對角線為對稱軸
    • 鷂形
  • 對角線均為對稱軸:
    • 菱形
  • 一條對稱軸:
    • 等腰梯形鹞形
  • 兩條對稱軸:
    • 矩形菱形
  • 四條對稱軸:
    • 正方形
  • 旋轉對稱重合兩次:
    • 平行四邊形
  • 旋轉對稱重合四次:
    • 正方形
  • 兩對對邊長度相等:
    • 平行四邊形
  • 兩對鄰邊長度相等:
    • 鷂形
  • 四邊長度相等:
    • 菱形正方形
  • 兩對對角相等:
    • 平行四邊形
  • 兩對相鄰角相等:
    • 等腰梯形
  • 四角相等:
    • 矩形
  • 對角和等於
    • 圓內接四邊形
  • 一對對邊平行:
    • 梯形
  • 兩對對邊平行:
    • 平行四邊形
  • 四邊可接圓形:
    • 圓外切四邊形
  • 两对边长度和相等:
    • 圆外切四邊形
  • 頂點都在一個圓上:
    • 圓內接四邊形

面積

任意凸[13]四邊形面積可以利用下列算式算出:[13]

其中表示兩對角線的長度,是對角線的夾角[14]正交四边形(如菱形、正方形或鷂形等),這個式子可以化簡成:

其中由於是90°,因此修正項可以消掉。

若凸四邊形的四邊長度分別是,對角線長度為,對角線相交的角度為,其面積為:

若對角線相交的角度為,四邊形的對邊的關係:

底下是一些針對特殊四邊形的面積公式:

扭歪四邊形

一種扭歪四邊形。

扭歪四邊形,又稱不共面四邊形,是指頂點並非完全共面的四邊形。因為扭歪四邊形不存在唯一確定的內部區域,故無法計算其面積。

參考文獻

  1. ^ 1.0 1.1 1.2 1.3 Euclid's Elements, Book I. mathcs.clarku.edu. [2017-10-21]. (原始内容存档于2017-09-18). 
  2. ^ Archived copy (PDF). [2013-06-20]. (原始内容 (PDF)存档于2014-05-14). 
  3. ^ Weisstein, Eric W. (编). Rhomboid. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  4. ^ Weisstein, Eric W. (编). Rhombus. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  5. ^ Colebrooke, Henry-Thomas, Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhascara, John Murray: 58, 1817 
  6. ^ Radic, Mirko; Kaliman, Zoran and Kadum, Vladimir, "A condition that a tangential quadrilateral is also a chordal one", Mathematical Communications, 12 (2007) pp. 33–52.
  7. ^ Bogomolny, Alexander. "Inscriptible and Exscriptible Quadrilaterals", Interactive Mathematics Miscellany and Puzzles. [2011-08-18]. (原始内容存档于2011-09-06). 
  8. ^ G. Keady, P. Scales and S. Z. Németh, "Watt Linkages and Quadrilaterals", The Mathematical Gazette Vol. 88, No. 513 (Nov., 2004), pp. 475–492.
  9. ^ A. K. Jobbings, "Quadric Quadrilaterals", The Mathematical Gazette Vol. 81, No. 491 (Jul., 1997), pp. 220–224.
  10. ^ R. A. Beauregard, "Diametric Quadrilaterals with Two Equal Sides", College Mathematics Journal Vol. 40, No. 1 (Jan 2009), pp. 17-21.
  11. ^ Stars: A Second Look (PDF). [2016-08-25]. (原始内容 (PDF)存档于2016-03-03). 
  12. ^ Quadrilaterals. technologyuk. [2016-08-25]. (原始内容存档于2017-07-06). 
  13. ^ 13.0 13.1 Weisstein, Eric W. (编). Quadrilateral. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. (英语). 
  14. ^ Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.

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