ヘロンの公式
ヘロンの公式 ― 3辺の長さが a, b, c である三角形の面積 S は
S = s ( s − a ) ( s − b ) ( s − c ) {\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}
ただし
s = a + b + c 2 {\displaystyle s={\frac {a+b+c}{2}}}
また、以下のような s を用いない表記もある。
S = ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) 4 {\displaystyle S={\frac {\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}{4}}}
S = 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) − ( a 4 + b 4 + c 4 ) 4 {\displaystyle S={\frac {\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}{4}}}
S = ( a 2 + b 2 + c 2 ) 2 − 2 ( a 4 + b 4 + c 4 ) 4 {\displaystyle S={\frac {\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}{4}}}
S = 4 b 2 c 2 − ( a 2 − b 2 − c 2 ) 2 4 {\displaystyle S={\frac {\sqrt {4b^{2}c^{2}-(a^{2}-b^{2}-c^{2})^{2}}}{4}}}
S = a h 2 = a b 2 sin  C = a b 2 1 − cos 2  C = a b 2 ( 1 + cos  C ) ( 1 − cos  C ) = a b 2 ( 1 + a 2 + b 2 − c 2 2 a b ) ( 1 − a 2 + b 2 − c 2 2 a b ) = a b 2 a 2 + 2 a b + b 2 − c 2 2 a b ⋅ − ( a 2 − 2 a b + b 2 − c 2 ) 2 a b = ( a + b ) 2 − c 2 4 ⋅ − [ ( a − b ) 2 − c 2 ] 4 = ( a + b + c ) ( a + b − c ) 4 ⋅ − ( a − b + c ) ( a − b − c ) 4 = a + b + c 2 ⋅ − a + b + c 2 ⋅ a − b + c 2 ⋅ a + b − c 2 = a + b + c 2 ( a + b + c 2 − a ) ( a + b + c 2 − b ) ( a + b + c 2 − c ) {\displaystyle {\begin{aligned}S&={\frac {ah}{2}}={\frac {ab}{2}}\sin C={\frac {ab}{2}}{\sqrt {1-\cos ^{2}C}}\\&={\frac {ab}{2}}{\sqrt {(1+\cos C)(1-\cos C)}}\\&={\frac {ab}{2}}{\sqrt {\left(1+{\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)\left(1-{\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)}}\\&={\frac {ab}{2}}{\sqrt {{\frac {a^{2}+2ab+b^{2}-c^{2}}{2ab}}\cdot {\frac {-(a^{2}-2ab+b^{2}-c^{2})}{2ab}}}}\\&={\sqrt {{\frac {(a+b)^{2}-c^{2}}{4}}\cdot {\frac {-[(a-b)^{2}-c^{2}]}{4}}}}\\&={\sqrt {{\frac {(a+b+c)(a+b-c)}{4}}\cdot {\frac {-(a-b+c)(a-b-c)}{4}}}}\\&={\sqrt {{\frac {a+b+c}{2}}\cdot {\frac {-a+b+c}{2}}\cdot {\frac {a-b+c}{2}}\cdot {\frac {a+b-c}{2}}}}\\&={\sqrt {{\frac {a+b+c}{2}}\left({\frac {a+b+c}{2}}-a\right)\left({\frac {a+b+c}{2}}-b\right)\left({\frac {a+b+c}{2}}-c\right)}}\end{aligned}}}