Python 2.7 306 Bytes
import math as z,re
t,m,r,w=float,map,reduce,[e.split() for e in re.split(' (?=[a-z])| (?=\d+(?:\.\d+)?$)',raw_input())]
def f(S,D):i=r(lambda x,y:x*y,D);return((i,i*.5)[S[0]=='t'],3.14*i*D[0])[S[0]=="c"]
print z.ceil(r(lambda x,y:x/y,m(lambda q:f(q[0],q[1:]),m(lambda x:[x[0]]+m(t,x[1:]),w[:-1]))))*t(*w[-1])
Takes input from stdin.
Testing it-
$ python pool.py
triangle 10 12.25 3 circle 5 2.2 5
10.0
$ python pool.py
triangle 5 87.3 20001 rectangle 5.14 2 105.623 0.2
804.2