36 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
Join any vertex, as B, with all of the other vertices of the bases. The 
prismatoid is composed of pyramid, B—EGF (having base in upper base 
of prismatoid); the pyramids E—BDC, E—ADB (having their base in 
the lower base of prismatoid; and the tetrahedra, GE—BC and AB—FE. 
There are thus three types of solids that make up the prismatoid. We 
shall first show that Koppe’s theorem applies to each class separately. 
Given V=MH=H^^— \ A) 
Where M=mean area of prismatoid. 
B=area lower base. 
C=area upper base. 
A=area base of associate pyramid. 
E 
(a) In B—EGF we have 
V=^HC, 
But by associate formula, 
V=H 
B±C_ 
2 '« -J 
C 
=H(0+2— *C) = *HC. 
(b) In FE—BC we have 
V=-g( B-f- C+4 SyJ ==-g( 0 -f 0 + 4Si /2 )= f HS14 , 
But by associate formula 
V=H 
-H(0+0+|S !i ,)=fHS i4 . 
From Fig. 3 (b) we see that 
BCl2=4(abcd), 
Or A=4Si/. 
