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,
AD=BC=r,
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