The base of a triangular pyramid is a triangle with corners at #(5 ,1 )#, #(2 ,3 )#, and #(3 ,4 )#. If the pyramid has a height of #4 #, what is the pyramid's volume?

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Answer 1 · 2017-06-30T06:11:48

The volume is #=3.33u^3#

The area of the base triangle is calculated with the determinant

Therefore, the area of the base is

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

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

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

#=1/2(5(3-4)-1(2-3)+1(8-9))#

#=1/2(-5+1-1)#

#=1/2|-5|=5/2#

The volume of the pyramid is

#=1/3*h*A=1/3*4*5/2=10/3=3.33u^3#