The base of a triangular pyramid is a triangle with corners at #(1 ,1 )#, #(3 ,6 )#, and #(4 ,5 )#. If the pyramid has a height of #5 #, what is the pyramid's volume?

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Answer 1 · 2017-07-12T13:47:38

The volume is #=5.83u^3#

The area of the base is

#A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

#=1/2(x_1(y_2-y_3)-y_1(x_2-x_3)+(x_2y_3-x_3y_2))#

#A=1/2|(1,1,1),(3,6,1),(4,5,1)|#

#=1/2(1*|(6,1),(5,1)|-1*|(3,1),(4,1)|+1*|(3,6),(4,5)|)#

#=1/2(1(6-5)-1(3-4)+1(15-24))#

#=1/2(1+1-9)#

#=1/2|-7|=7/2#

The volume of the pyramid is

#V=1/3*A*h#

#=1/3*7/2*5#

#=35/6#

#=5.83#