ヘロンの公式
h 2 = 2 s ( 2 s − 2 b ) ( 2 s − 2 c ) ( 2 s − 2 a ) 4 a 2 = 16 s ( s − a ) ( s − b ) ( s − c ) 4 a 2 {\displaystyle {\begin{aligned}h^{2}&={\frac {2s(2s-2b)(2s-2c)(2s-2a)}{4a^{2}}}\\&={\frac {16s(s-a)(s-b)(s-c)}{4a^{2}}}\end{aligned}}}
h = 2 s ( s − a ) ( s − b ) ( s − c ) a {\displaystyle {\begin{aligned}h&={\frac {2{\sqrt {s(s-a)(s-b)(s-c)}}}{a}}\end{aligned}}}
よって
S = 1 2 a h = 1 2 a 2 s ( s − a ) ( s − b ) ( s − c ) a = s ( s − a ) ( s − b ) ( s − c ) {\displaystyle {\begin{aligned}S&={{\text{1}} \over {\text{2}}}ah\\&={{\text{1}} \over {\text{2}}}a{\frac {2{\sqrt {s(s-a)(s-b)(s-c)}}}{a}}\\&={\sqrt {s(s-a)(s-b)(s-c)}}\end{aligned}}}
が得られる。
{\displaystyle {\begin{aligned}V^{2}=&{\frac {1}{144}}\{{l_{1}}^{2}{l_{5}}^{2}(-{l_{1}}^{2}+{l_{2}}^{2}+{l_{3}}^{2}+{l_{4}}^{2}-{l_{5}}^{2}+{l_{6}}^{2})+{l_{2}}^{2}{l_{6}}^{2}({l_{1}}^{2}-{l_{2}}^{2}+{l_{3}}^{2}+{l_{4}}^{2}+{l_{5}}^{2}-{l_{6}}^{2})\\&+{l_{3}}^{2}{l_{4}}^{2}({l_{1}}^{2}+{l_{2}}^{2}-{l_{3}}^{2}-{l_{4}}^{2}+{l_{5}}^{2}+{l_{6}}^{2})-{l_{1}}^{2}{l_{2}}^{2}{l_{4}}^{2}-{l_{2}}^{2}{l_{3}}^{2}{l_{5}}^{2}-{l_{1}}^{2}{l_{3}}^{2}{l_{6}}^{2}-{l_{4}}^{2}{l_{5}}^{2}{l_{6}}^{2}\}.\end{aligned}}}
ただし、四面体の頂点を O, A, B, C とすると、 l 1 = | OA | {\displaystyle l_{1}=|{\text{OA}}|} , l 2 = | OB | {\displaystyle l_{2}=|{\text{OB}}|} , l 3 = | OC | {\displaystyle l_{3}=|{\text{OC}}|} , l 4 = | AB | {\displaystyle l_{4}=|{\text{AB}}|} , l 5 = | BC | {\displaystyle l_{5}=|{\text{BC}}|} , l 6 = | CA | {\displaystyle l_{6}=|{\text{CA}}|} .