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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?

 Submitted by Federico Kereki Rating: 4.1250 (8 votes) Solution: (Hide) The volume of such a pyramid is zero; the four points are coplanar. To see why, let's work with coordinates. Let A be at (0,0,0), B at (p,0,0), C at (0,q,0), and D at (x,y,z).We have AD=r, so x²+y²+z²=r²; BD=q, so (x-p)²+y²+z²=q²; and CD=p, so x²+(y-q)²+z²=p².Subtracting BD from AD, 2px-x²=p², so x=p. Subtracting CD from AD, 2qy-q²=q², so y=q. Finally, substituting these values in AD, p²+q²+z²=r², so z=0, and D lies in the same plane as A, B and C.

 Subject Author Date Slaying the Hedra (Spoiler) Dej Mar 2007-12-06 11:17:48 answer K Sengupta 2007-05-04 14:53:58 No Subject nikki 2004-04-28 22:47:20 re: Solution Charlie 2004-04-27 09:18:39 re: Solution Charlie 2004-04-26 20:58:03 re: Solution Charlie 2004-04-26 20:55:42 re: Solution Oskar 2004-04-26 17:46:50 Solution Victor Zapana 2004-04-26 16:59:56 Solution by rotation e.g. 2004-04-26 15:42:41 Working with spheres e.g. 2004-04-26 12:25:38 Another View Jer 2004-04-26 12:11:38 Geometrically Oskar 2004-04-26 10:27:19 Two ways Charlie 2004-04-26 09:51:06 Is this true? Oskar 2004-04-26 09:28:51 Strange... e.g. 2004-04-26 09:25:34

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