 Lv 748,677 points

# Fred

Lifelong interests in math, physics, astronomy, music, including performing (amateur). Adult lifetime interests in running, bicycling. I have benefited from some excellent tutelage in mathematics and physics, and would like to pay that forward in whatever small measure possible.

• ### What is Heron's formula in n dimensions?

In plane geometry, knowing all the side lengths of a polygon of m sides isn't enough, by itself, to determine the area; unless m=3, in which case there is Heron's formula:

A = √[s(s - a)(s - b)(s - c)], where s (the semiperimeter) = ½(a + b + c)

In solid geometry, a similar situation exists for the tetrahedron -- knowing all 6 edge lengths is sufficient to completely determine its shape, and thus, its volume.

A) What is the Heron-like formula for that?

B) What is the formula for a simplex (hyperpyramid) in n dimensions, given all ½n(n+1) edge lengths?

Unlike my first Y!A question, I don't have prior knowledge of the answer to this.

And elegance/simplicity in the final expression will get extra consideration.