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 Pythagorean Pyramid (Posted on 2004-04-26)
The pictured tetrahedron has four identical rectangular (i.e., right-angled or pythagorean) triangles as faces, with

AB=CD=p,
AC=BD=q,
and p²+q²=r².

What's its volume, as a function of p, q and r?

 See The Solution Submitted by Federico Kereki Rating: 4.1250 (8 votes)

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 re: Solution | Comment 10 of 15 |
(In reply to Solution by Victor Zapana)

`In the preceding discussion we developed a tetrahedral version of Heron's formula for a restricted class of tetrahedra, namely thosethat can serve as the hypotenuse of a "right" 4D simplex, but thereare other special classes of tetrahedra that possess interestingvolume formulas.  The one that gives the closest analogue to Heron'sformula is the class of tetrahedra whose opposite edges lengthsare equal.  Thus there are only three independent edge lengths, andeach face of the tetrahedron is identical.  Letting (a,f), (b,e), and (c,d) denote the pairs of opposite edge lengths, we can set a=f, b=e,and c=d in the basic determinant expression for the volume, or equivalently in Piero della Francesca's formula, and we find thatthe resulting expression for the squared volume factors as   72 V^2  =  (-a^2 + b^2 + c^2)(a^2 - b^2 + c^2)(a^2 + b^2 - c^2)which is certainly reminiscent of Heron's formula for the area ofeach face           16 A^2  =  (a+b+c)(-a+b+c)(a-b+c)(a+b-c)This also shows that if each face is an identical right triangle, thevolume is zero, as it must be, since four such triangles connectedby their edges to give a tetrahedron necessarily all lie flat in thesame plane:              ________             |\      /|             | \    / |             |  \  /  |             |   \/   |             |   /\   |             |  /  \  |             | /    \ |             |/______\|Obviously we can construct a regular tetrahedron with equilateraltriangles of the same area as these right triangles, and the volumeis V = a^3 / sqrt(72), which illustrates the fact that the face areas of a tetrahedron do not in general determine it's volume.`

 Posted by Charlie on 2004-04-26 20:55:42

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