Nice

Niccolo Fontana Tartaglia, whom you may recognise from having discovered Cardano’s solution to the general cubic equation, also discovered a generalisation of Heron’s formula to compute the volume of a tetrahedron:

As you may expect, this can be generalised to compute the volume of any *n*-simplex (*n* = 2 reducing to Heron’s formula for the area of a triangle). I wondered how one would go about proving this identity, and then realised it can be accomplished by elementary facts about determinants. Firstly, it is easy to show the following result:

- The volume of an
*n*-simplex*S*with vertices at {,**e**_1, …,**e**_*n*} is equal to 1/*n*!, where**e**_*i*is the*i*th standard basis vector.

This can be proved, for instance, by subdividing a unit cube into *n*! simplices, each of which is congruent to *S*. Now…

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