I'm reading the wikipedia page https://en.m.wikipedia.org/wiki/Menger_curvature

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It mensions a so-called well-known formula here but doesn't mention its name. I guess it is heron's formula. But again it is very different from this formula. Heron's formula is $A=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\dfrac{a+b+c}2$. But it contains a square root and has no relationship with the radius R. I wonder which formula of the radius R it uses here to deduce this formula of curvature?

  • $\begingroup$ Remember $4RK=abc$ where $K=rs$ is the area of triangle ABC. $\endgroup$ Commented Sep 1, 2023 at 6:16
  • $\begingroup$ @user10354138 Thank you. Which formula is this? $\endgroup$
    – user900476
    Commented Sep 1, 2023 at 6:24
  • $\begingroup$ Eliminate $\sin C$ from the well-known formula $K=\frac12 ab\sin C$ and $c=2R\sin C$. That is how you get $4Rrs=abc$ relating all six important lengths of a Euclidean triangle. $\endgroup$ Commented Sep 1, 2023 at 6:28
  • $\begingroup$ @user10354138 Thank you! I didn't realize it uses this formula. I kind of forgot $A=\dfrac12 ab\sin C$ $\endgroup$
    – user900476
    Commented Sep 1, 2023 at 6:35


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