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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
Volume of a Pyramid
Posted: Jan 21, 1993 12:36 PM

Newsgroups: sci.math
From: orourke@sophia.smith.edu (Joseph O'Rourke)
Subject: Re: pyramid volume
Message-ID: <1993Jan21.173423.11339@sophia.smith.edu>
Organization: Smith College, Northampton, MA, US
References: <1993Jan21.140402.25519@mr.med.ge.com>
Date: Thu, 21 Jan 1993 17:34:23 GMT

In article <1993Jan21.140402.25519@mr.med.ge.com> carl@crazyman.med.ge.com (Carl Crawford) writes:
>
>how do show that the volume of a pyramid is
>
> 1/3 * area of base * altitude
>
>without using calculus?

Nice question! Although I don't have an answer, I would like to
make an observation: three identical tetrahedra pack half a cube.
Take the unit cube
0 <= x <= 1
0 <= y <= 1
0 <= z <= 1
and intersect it with the halfspace x+y <= 1. The result is a
unit-height triangular prism, the convex hull of
000, 100, 010,
001, 101, 011,
where "000" means "(0,0,0)" etc. Suppose you believe the volume of
this is V = A * h = A, where A is the base area and h=1 the height.
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.
T1 has base A on the plane z=0, and height 1;
T2 has base A on the plane z=1, and height 1;
T3 has base A on the plane x=0, and height 1.
Since these three are congruent, each has volume V/3.
I looked for this construction in several books without luck.

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