Hero's formula for the area A of a triangle in terms of its edge lengths u,v,w can be written as a product 16 A^2 = (+u+v+w)(-u+v+w)(+u-v+w)(+u+v-w); a triangle is "Heronian" when u,v,w are integers and A is rational, in which case it can be shown that A will be an integer multiple of 6; exercising a modicum of ingenuity, this may be proved without resort to assistance from a computer. Furthermore, the polynomial factorisation makes it clear that A must be expected to be "rounder" --- that is, have more small prime factors --- than an integer of comparable size chosen at random. For a tetrahedron, the volume V in terms of edge lengths u,v,w,x,y,z is given by 288 V^2 = CMD_3(u,...,z), where CMD_n denotes the order-(n+2) Cayley-Menger determinant, a polynomial cubic in u^2,...,z^2 with (even) integer coefficients. Notice that for n > 2, these polynomials are irreducible. See http://en.wikipedia.org/wiki/Distance_geometry . A tetrahedron is "Heronian" when the lengths are integer, and all four A's and V are rational: V will be an integer multiple of 84, and almost certainly of 168. Contrary to several recent claims --- names being withheld to protect the guilty --- proof of the latter result currently appears to involve a computation of 500--1000 (elderly laptop) hours. In the course of attempting to find a less onerous (not to mention more illuminating) proof of such volume divisibility properties, I was led to factorise the volumes of the known sample Heronian tetrahedra, as listed at http://www.wm.uni-bayreuth.de/fileadmin/Sascha/Research/Heronian/Primitive_H... . The results were mildly astonishing: for example, number 170 with diameter 5920 is the smallest for which any prime factor of its volume exceeds 100. [A random integer of this size would have some prime factor of order 10^7.] At the same time, I looked at the gradient vector [d/du,...,d/dz] CMD_3 --- and the same phenomenon is to a large extent repeated, albeit with somewhat larger primes occasionally intruding. Sample case #49 : edge lengths = [ 1480, 1275, 1189, 1248, 1160, 261 ] ; 288 vol^2 = 2^19 * 3^10 * 7^2 * 29^4 ; grad(288 vol^2) = [ 2^19 * 3^5 * 5^1 * 29^3 * 37^1 , 0 , 0 , 2^15 * 3^5 * 13^2 * 29^3 * 41^1 , 2^17 * 3^5 * 5^1 * 17^1 * 23^1 * 29^3 , 2^20 * 3^8 * 7^2 * 29^3 ]. So what's going on here: is there some theory that explains why these numbers are so round? The phenomenon does not appear to be a property of CMD_3 per se --- nonsquare values are not round. Is it a consequence of their squareness --- now that would be ironic! I haven't investigated whether the areas' rationality is relevant. Fred Lunnon