ヘロンの公式
) 2 | 0 d 12 2 ⋯ d 1 ( n + 1 ) 2 1 d 21 2 0 ⋯ d 2 ( n + 1 ) 2 1 ⋯ d ( n + 1 ) 1 2 d ( n + 1 ) 2 2 ⋯ 0 1 1 1 ⋯ 1 0 | {\displaystyle V^{2}={\frac {(-1)^{n+1}}{2^{n}(n!)^{2}}}\left|{\begin{array}{ccccc}0&{d_{12}}^{2}&\cdots &{d_{1(n+1)}}^{2}&1\\{d_{21}}^{2}&0&\cdots &{d_{2(n+1)}}^{2}&1\\&&\cdots &&\\{d_{(n+1)1}}^{2}&{d_{(n+1)2}}^{2}&\cdots &0&1\\1&1&\cdots &1&0\end{array}}\right|}
ただし、 d i j {\displaystyle d_{ij}} は頂点 i = 1 , 2 , ⋯ , n + 1 {\displaystyle i=1,2,\cdots ,n+1} と頂点 j = 1 , 2 , ⋯ , n + 1 {\displaystyle j=1,2,\cdots ,n+1} を結ぶ辺の長さ。