These results are neither obvious, nor known by heart (except by a few genius!); it is thus suitable to verify them.
We can compute the volume again by using the cube in which the tetrahedron is naturally inscribed.
The diagonal of a face is equal to a (edge of the tetrahedron), thus the edge of the cube is c=a/√2.
To get the tetrahedron we have to cut off four "corners" of the cube (trirectangular tetrahedra with equilateral base). If we use one of the isoscelesrectangular face as base, the hight of a "corner" is c, and its volume (1/3)×(c²/2)×c = c³/6
we deduce the tetrahedron's volume: c³4(c³/6) = c³/3 = (a/√2)³/3 = (√2/12)×a³
Corollary 1: the hight of the trirectangular tetrahedron relative to the equilateral face is the third of the cube's diagonal (d=c√3).
From V = (1/3)×A×h we deduce h = 3V/A = 3[c³/6]/[(1/2)×(c√2)×(c√2)(√3/2)] = c/√3 = c√3/3 = d/3
Corollary 2: The circumsphere (defined by the four vertices) is, by symmetry, centered at the intersection point of the altitudes; its center O lays at the quarter of each altitude (starting from the base), its radius is thus R = (3/4)×h.
proofs: • O is the isobarycentre of the vertices O = bar[A(1),B(1),C(1),S(1)] = bar[bar[A(1),B(1),C(1)],S(1)] = bar[H(3),S(1)] situated at the quarter of HS
• One may also use the cube which has four vertices which define the tetrahedron; the two polyhedra have same circumsphere.
An altitude of the tetrahedron lays on a diagonal of the cube and h=(2/3)d thus R = d/2 = 1/2×(3/2)h = (3/4)×h
verification: a=c√2, d=c√3 and h=(2/3)d lead to R = d/2 = (3h/2)/2 = (3/4)×h or R = (c√3)/2 = ((a/√2)√3)/2 = a×(√6/4) = a(√2/√3)×(3/4) = h×(3/4)
remarks: radius of the insphere (tangent to the faces, center O) : r = h/4 = R/3 = a×(√6/12)
radius of the midsphere (tangent to the edges, center O): by Pythagoras theorem ρ² = R²(a/2)² which gives ρ = a(√2/4)
Since we don't see how to avoid trigonometry to compute the angle, a verification by computation would not be very convincing. Nevertheless we can proceed to an experimental verification with what's available. We "recover" the angle of two faces between the arms of a compass and then we measure it with a protractor; it's just an approximate verification, but one can reach an outstanding precision taking into account the rudimentary means used.
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