鎬庝箞姹備笁瑙掑舰鐨勯潰绉?/h1>
涓€銆佹€庝箞姹備笁瑙掑舰鐨勯潰绉?/h2>
娴蜂鸡鍏紡鍙堣瘧甯屼鸡鍏紡锛屼紶璇存槸鍙や唬鐨勫彊鎷夊彜鍥界帇甯屼鸡浜屼笘鍙戠幇鐨勫叕寮忥紝鍒╃敤涓夎褰㈢殑涓夋潯杈归暱鏉ユ眰鍙栦笁瑙掑舰闈㈢Н銆備絾鏍规嵁Morris Kline鍦?908骞村嚭鐗堢殑鐫€浣滆€冭瘉锛岃繖鏉″叕寮忓叾瀹炴槸闃垮熀绫冲痉鎵€鍙戠幇锛屼互鎵樺笇浼︿簩涓栫殑鍚嶅彂琛ㄣ€
鍋囪鏈変竴涓笁瑙掑舰锛岃竟闀垮垎鍒负a銆乥銆乧锛屼笁瑙掑舰鐨勯潰绉疭鍙敱浠ヤ笅鍏紡姹傚緱锛
S=sqrt{s(s-a)(s-b)(s-c)}
鑰屽叕寮忛噷鐨剆锛
s=frac{a+b+c}{2}
鐢变簬浠讳綍n杈圭殑澶氳竟褰㈤兘鍙互鍒嗗壊鎴恘-2涓笁瑙掑舰锛屾墍浠ユ捣浼﹀叕寮忓彲浠ョ敤浣滄眰澶氳竟褰㈤潰绉殑鍏紡銆傛瘮濡傝娴嬮噺鍦熷湴鐨勯潰绉殑鏃跺€欙紝涓嶇敤娴嬩笁瑙掑舰鐨勯珮锛屽彧闇€娴嬩袱鐐归棿鐨勮窛绂伙紝灏卞彲浠ユ柟渚垮湴瀵煎嚭绛旀銆
[缂栬緫]璇佹槑
涓庢捣浼﹀湪浠栫殑鐫€浣淢etrica涓殑鍘熷璇佹槑涓嶅悓锛屽湪姝ゆ垜浠敤涓夎鍏紡鍜屽叕寮忓彉褰㈡潵璇佹槑銆傝涓夎褰㈢殑涓夎竟a銆乥銆乧鐨勫瑙掑垎鍒负A銆丅銆丆锛屽垯棣€寮﹀畾鐞嗕负
cos(C) = frac{a^2+b^2-c^2}{2ab}
浠庤€屾湁
sin(C) = sqrt{1-cos^2(C)} = frac{ sqrt{-a^4 -b^4 -c^4 +2a^2b^2 +2b^2c^2 +2c^2a^2} }{2ab}
鍥犳涓夎褰㈢殑闈㈢НS涓
S = frac{1}{2}ab sin(C)
= frac{1}{4}sqrt{-a^4 -b^4 -c^4 +2a^2b^2 +2b^2c^2 +2c^2a^2}
= sqrt{s(s-a)(s-b)(s-c)}
鏈€鍚庣殑绛夊彿閮ㄥ垎鍙敤鍥犲紡鍒嗚В浜堜互瀵煎嚭銆
[缂栬緫]澶栭儴杩炵粨
棣欐腐绉戞妧澶у鏁板绯胡鏁板鏁版嵁搴掹闃垮熀绫冲痉鐨勬暟瀛︽垚灏卞拰鐮旂┒鏂规硶
浜屻€佽闂暟瀛︼細 涓夎褰紝瀹冪殑闈㈢Н璁＄畻鍏紡鏄€庢牱璁＄畻鐨勫憿锛 鏄笉鏄彲浠ヨ鏄袱涓笁瑙掑舰鎷稽/h2>
濡傚浘鎵€绀国
涓や釜瀹屽叏涓€鏍风殑涓夎褰㈤兘鍙互鎷兼垚涓€涓钩琛屽洓杈瑰舰锛屾嫾鎴愮殑骞宠鍥涜竟褰㈢殑闈㈢Н绛変簬杩欎袱涓笁瑙掑舰鐨勯潰绉箣鍜岋紝搴曠瓑浜庝笁瑙掑舰鐨勫簳锛岄珮绛変簬涓夎褰㈢殑楂橈紝鎵€浠ヤ竴涓笁瑙掑舰鐨勯潰绉?杩欎釜骞宠鍥涜竟褰㈢殑闈㈢Н鐨勪竴鍗婏紝鍥犱负骞宠鍥涜竟褰㈢殑闈㈢Н=搴暶楅珮锛屼笁瑙掑舰鐨勯潰绉?=搴暶楅珮銆備袱涓畬鍏ㄤ竴鏍风殑涓夎褰㈤兘鍙互鎷兼垚涓€涓钩琛屽洓杈瑰舰锛屾嫾鎴愮殑骞宠鍥涜竟褰㈢殑闈㈢Н绛変簬杩欎袱涓笁瑙掑舰鐨勯潰绉箣鍜岋紝搴曠瓑浜庝笁瑙掑舰鐨勫簳锛岄珮绛変簬涓夎褰㈢殑楂橈紝
鎵€浠ヤ竴涓笁瑙掑舰鐨勯潰绉?杩欎釜骞宠鍥涜竟褰㈢殑闈㈢Н鐨勪竴鍗婏紝鍥犱负骞宠鍥涜竟褰㈢殑闈㈢Н=搴暶楅珮锛屼笁瑙掑舰鐨勯潰绉?=搴暶楅珮銆侟/p>
涓夈€佷笁瑙掑舰鐨勯潰绉叕寮忔槸浠€涔堬紵
浣犲ソ
涓夎褰㈤潰绉?搴?瀵瑰簲杈逛笂鐨勯珮/2
S=ah/2
杩欎釜搴曪紙杈癸級a鏄换鎰忓彇鐨勶紝
楂榟鍒欐槸鏍规嵁浣犲彇鐨勫簳鍋氬嚭鏉ョ殑锛屼粠鍙﹀涓€涓《鐐瑰埌杩欐潯杈 a鐨勫瀭绾挎闀裤€侟/p>
鍥涖€佷笁瑙掑舰闈㈢Н濡備綍璁＄畻锛烖/h2>
涓夎褰㈢殑闈㈢Н鍏紡
(1)S鈻昌1/2ah 锛坅鏄笁瑙掑舰鐨勫簳锛宧鏄簳鎵€瀵瑰簲鐨勯珮锛 (2)S鈻昌1/2acsinB锛?/2bcsinA锛?/2absinC 锛堜笁涓涓衡垹A鈭燘鈭燙锛屽杈瑰垎鍒负a,b,c锛屽弬瑙佷笁瑙掑嚱鏁帮級 (3)S鈻昌鈭氥€攑(p-a)(p-b)(p-c)銆 銆攑=1/2(a+b+c)銆曪紙娴蜂鸡鈥旂Е涔濋煻鍏紡锛 (4)S鈻昌abc/(4R) (R鏄鎺ュ渾鍗婂緞) (5)S鈻昌1/2(a+b+c)r (r鏄唴鍒囧渾鍗婂緞) (6) ........... | a b 1 | S鈻昌1/2 | c d 1 | ............| e f 1 | 銆攟 a b 1 | ....| c d 1 | ....| e f 1 |涓轰笁闃惰鍒楀紡,姝や笁瑙掑舰ABC鍦ㄥ钩闈㈢洿瑙掑潗鏍囩郴鍐匒(a,b),B(c,d), C(e,f),杩欓噷ABC閫夊尯鍙栨渶濂芥寜閫嗘椂閽堥『搴忎粠鍙充笂瑙掑紑濮嬪彇锛屽洜涓鸿繖鏍峰彇寰楀嚭鐨勭粨鏋滀竴鑸兘涓烘鍊硷紝濡傛灉涓嶆寜杩欎釜瑙勫垯鍙栵紝鍙兘浼氬緱鍒拌礋鍊硷紝浣嗗彧瑕佸彇缁濆鍊煎氨鍙互浜嗭紝涓嶄細褰卞搷涓夎褰㈤潰绉殑澶у皬銆 (7)S鈻昌c^2sinAsinB/2sin(A+B) (8)S姝ｂ柍= [锛堚垰3锛?4]a^2 (姝ｄ笁瑙掑舰闈㈢Н鍏紡锛宎鏄笁瑙掑舰鐨勮竟闀?
鍩烘湰鍏ㄩ儴鐨勬柟娉曢兘缁欏嚭浜咟/p>
涓€銆佹€庝箞姹備笁瑙掑舰鐨勯潰绉?/h2>
娴蜂鸡鍏紡鍙堣瘧甯屼鸡鍏紡锛屼紶璇存槸鍙や唬鐨勫彊鎷夊彜鍥界帇甯屼鸡浜屼笘鍙戠幇鐨勫叕寮忥紝鍒╃敤涓夎褰㈢殑涓夋潯杈归暱鏉ユ眰鍙栦笁瑙掑舰闈㈢Н銆備絾鏍规嵁Morris Kline鍦?908骞村嚭鐗堢殑鐫€浣滆€冭瘉锛岃繖鏉″叕寮忓叾瀹炴槸闃垮熀绫冲痉鎵€鍙戠幇锛屼互鎵樺笇浼︿簩涓栫殑鍚嶅彂琛ㄣ€
鍋囪鏈変竴涓笁瑙掑舰锛岃竟闀垮垎鍒负a銆乥銆乧锛屼笁瑙掑舰鐨勯潰绉疭鍙敱浠ヤ笅鍏紡姹傚緱锛
S=sqrt{s(s-a)(s-b)(s-c)}
鑰屽叕寮忛噷鐨剆锛
s=frac{a+b+c}{2}
鐢变簬浠讳綍n杈圭殑澶氳竟褰㈤兘鍙互鍒嗗壊鎴恘-2涓笁瑙掑舰锛屾墍浠ユ捣浼﹀叕寮忓彲浠ョ敤浣滄眰澶氳竟褰㈤潰绉殑鍏紡銆傛瘮濡傝娴嬮噺鍦熷湴鐨勯潰绉殑鏃跺€欙紝涓嶇敤娴嬩笁瑙掑舰鐨勯珮锛屽彧闇€娴嬩袱鐐归棿鐨勮窛绂伙紝灏卞彲浠ユ柟渚垮湴瀵煎嚭绛旀銆
[缂栬緫]璇佹槑
涓庢捣浼﹀湪浠栫殑鐫€浣淢etrica涓殑鍘熷璇佹槑涓嶅悓锛屽湪姝ゆ垜浠敤涓夎鍏紡鍜屽叕寮忓彉褰㈡潵璇佹槑銆傝涓夎褰㈢殑涓夎竟a銆乥銆乧鐨勫瑙掑垎鍒负A銆丅銆丆锛屽垯棣€寮﹀畾鐞嗕负
cos(C) = frac{a^2+b^2-c^2}{2ab}
浠庤€屾湁
sin(C) = sqrt{1-cos^2(C)} = frac{ sqrt{-a^4 -b^4 -c^4 +2a^2b^2 +2b^2c^2 +2c^2a^2} }{2ab}
鍥犳涓夎褰㈢殑闈㈢НS涓
S = frac{1}{2}ab sin(C)
= frac{1}{4}sqrt{-a^4 -b^4 -c^4 +2a^2b^2 +2b^2c^2 +2c^2a^2}
= sqrt{s(s-a)(s-b)(s-c)}
鏈€鍚庣殑绛夊彿閮ㄥ垎鍙敤鍥犲紡鍒嗚В浜堜互瀵煎嚭銆
[缂栬緫]澶栭儴杩炵粨
棣欐腐绉戞妧澶у鏁板绯胡鏁板鏁版嵁搴掹闃垮熀绫冲痉鐨勬暟瀛︽垚灏卞拰鐮旂┒鏂规硶
浜屻€佽闂暟瀛︼細 涓夎褰紝瀹冪殑闈㈢Н璁＄畻鍏紡鏄€庢牱璁＄畻鐨勫憿锛 鏄笉鏄彲浠ヨ鏄袱涓笁瑙掑舰鎷稽/h2>
濡傚浘鎵€绀国
涓や釜瀹屽叏涓€鏍风殑涓夎褰㈤兘鍙互鎷兼垚涓€涓钩琛屽洓杈瑰舰锛屾嫾鎴愮殑骞宠鍥涜竟褰㈢殑闈㈢Н绛変簬杩欎袱涓笁瑙掑舰鐨勯潰绉箣鍜岋紝搴曠瓑浜庝笁瑙掑舰鐨勫簳锛岄珮绛変簬涓夎褰㈢殑楂橈紝鎵€浠ヤ竴涓笁瑙掑舰鐨勯潰绉?杩欎釜骞宠鍥涜竟褰㈢殑闈㈢Н鐨勪竴鍗婏紝鍥犱负骞宠鍥涜竟褰㈢殑闈㈢Н=搴暶楅珮锛屼笁瑙掑舰鐨勯潰绉?=搴暶楅珮銆備袱涓畬鍏ㄤ竴鏍风殑涓夎褰㈤兘鍙互鎷兼垚涓€涓钩琛屽洓杈瑰舰锛屾嫾鎴愮殑骞宠鍥涜竟褰㈢殑闈㈢Н绛変簬杩欎袱涓笁瑙掑舰鐨勯潰绉箣鍜岋紝搴曠瓑浜庝笁瑙掑舰鐨勫簳锛岄珮绛変簬涓夎褰㈢殑楂橈紝
鎵€浠ヤ竴涓笁瑙掑舰鐨勯潰绉?杩欎釜骞宠鍥涜竟褰㈢殑闈㈢Н鐨勪竴鍗婏紝鍥犱负骞宠鍥涜竟褰㈢殑闈㈢Н=搴暶楅珮锛屼笁瑙掑舰鐨勯潰绉?=搴暶楅珮銆侟/p>
涓夈€佷笁瑙掑舰鐨勯潰绉叕寮忔槸浠€涔堬紵
浣犲ソ
涓夎褰㈤潰绉?搴?瀵瑰簲杈逛笂鐨勯珮/2
S=ah/2
杩欎釜搴曪紙杈癸級a鏄换鎰忓彇鐨勶紝
楂榟鍒欐槸鏍规嵁浣犲彇鐨勫簳鍋氬嚭鏉ョ殑锛屼粠鍙﹀涓€涓《鐐瑰埌杩欐潯杈 a鐨勫瀭绾挎闀裤€侟/p>
鍥涖€佷笁瑙掑舰闈㈢Н濡備綍璁＄畻锛烖/h2>
涓夎褰㈢殑闈㈢Н鍏紡
(1)S鈻昌1/2ah 锛坅鏄笁瑙掑舰鐨勫簳锛宧鏄簳鎵€瀵瑰簲鐨勯珮锛 (2)S鈻昌1/2acsinB锛?/2bcsinA锛?/2absinC 锛堜笁涓涓衡垹A鈭燘鈭燙锛屽杈瑰垎鍒负a,b,c锛屽弬瑙佷笁瑙掑嚱鏁帮級 (3)S鈻昌鈭氥€攑(p-a)(p-b)(p-c)銆 銆攑=1/2(a+b+c)銆曪紙娴蜂鸡鈥旂Е涔濋煻鍏紡锛 (4)S鈻昌abc/(4R) (R鏄鎺ュ渾鍗婂緞) (5)S鈻昌1/2(a+b+c)r (r鏄唴鍒囧渾鍗婂緞) (6) ........... | a b 1 | S鈻昌1/2 | c d 1 | ............| e f 1 | 銆攟 a b 1 | ....| c d 1 | ....| e f 1 |涓轰笁闃惰鍒楀紡,姝や笁瑙掑舰ABC鍦ㄥ钩闈㈢洿瑙掑潗鏍囩郴鍐匒(a,b),B(c,d), C(e,f),杩欓噷ABC閫夊尯鍙栨渶濂芥寜閫嗘椂閽堥『搴忎粠鍙充笂瑙掑紑濮嬪彇锛屽洜涓鸿繖鏍峰彇寰楀嚭鐨勭粨鏋滀竴鑸兘涓烘鍊硷紝濡傛灉涓嶆寜杩欎釜瑙勫垯鍙栵紝鍙兘浼氬緱鍒拌礋鍊硷紝浣嗗彧瑕佸彇缁濆鍊煎氨鍙互浜嗭紝涓嶄細褰卞搷涓夎褰㈤潰绉殑澶у皬銆 (7)S鈻昌c^2sinAsinB/2sin(A+B) (8)S姝ｂ柍= [锛堚垰3锛?4]a^2 (姝ｄ笁瑙掑舰闈㈢Н鍏紡锛宎鏄笁瑙掑舰鐨勮竟闀?
鍩烘湰鍏ㄩ儴鐨勬柟娉曢兘缁欏嚭浜咟/p>
