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Topic: Volume of a Pyramid
Replies: 4   Last Post: Jan 29, 1993 12:01 AM

 Messages: [ Previous | Next ]
 Joseph O'Rourke Posts: 38 Registered: 12/3/04
Re: Volume of pyramid
Posted: Jan 22, 1993 3:19 PM

From: orourke@sophia.smith.edu (Joseph O'Rourke) | |-(1)--(1)--(1)
[1] Re: pyramid volume | |-(1)
Organization: Smith College, Northampton, MA, US | \-[1]
Date: Fri Jan 22 15:17:30 EST 1993 |-(1)
+ \-(1)--(1)

In article <1993Jan21.173423.11339@sophia.smith.edu> I wrote:
>[...] a unit-height triangular prism, the convex hull of
> 000, 100, 010,
> 001, 101, 011,
>where "000" means "(0,0,0)" etc. [...]
>Now partition the prism into three right tetrahedra, the hulls of
> T1: 000, 100, 010, 001;
> T2: 001, 101, 011, 100;
> T3: 000, 001, 011, 100.
>[...] Since these three are congruent, each has volume V/3.

^^^^^^^^^
Nick Haines (nickh@cmu.edu) pointed out to me that these are *not*
congruent, although they all have congruent bases and the same height.
Thus one needs the fact that two tetrahedra with congruent bases
and the same height have the same volume. Euclid establishes a
stronger version of this in XII.5.

Date Subject Author
1/21/93 Joseph O'Rourke
1/21/93 Joe Malkevitch
1/22/93 Joseph O'Rourke
1/22/93 Joseph O'Rourke
1/29/93 James King