The base of a triangular pyramid is a triangle with corners at #(6 ,5 )#, #(2 ,2 )#, and #(4 ,7 )#. If the pyramid has a height of #8 #, what is the pyramid's volume?

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Answer 1 · 2016-11-20T04:01:12

Volume #V=18 2/3=18.67" "#cubic units

To solve for the volume of the pyramid which is
#V=1/3*area*height#

Solve for the area of the triangle first with #P_1(2, 2),P_2(6, 5),P_3(4, 7)#

the area A matrix

#A=1/2*[(x_1,x_2,x_3,x_1),(y_1,y_2,y_3,y_1)]#

#A=1/2*(x_1*y_2+x_2*y_3+x_3*y_1-x_2*y_1-x_3*y_2-x_1*y_3)#

#A=1/2*[(2,6,4,2),(2,5,7,2)]#

#A=1/2*[2(5)+6(7)+4(2)-6(2)-4(5)-2(7)]#

#A=1/2*(10+42+8-12-20-14)#

#A=1/2*(60-46)#

#A=7#

Now the volume V

#V=1/3*area*height#

#V=1/3*7*8=56/3#

#V=18 2/3#