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

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Answer 1 · 2018-05-05T16:56:54

#7" units"^3#

the volume of a pyramid is found by the formula

#V=1/3xx" base area "xx" height"#

so the real problem is finding the area of the base

we are given that the base is a triangle and its vertices are given as co-ordinates

for a triangle with co-cordinates

#(x_1,y_1),(x_2,y_2),(x_3,y_3)#

the area can be calculated by the determinant

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

we ahve therefore

#A=+-1/2|(6,7,1),(5,5,1),(8,4,1)|#

#R'_1=R_1-R_2#

#A=+-1/2|(1,2,0),(5,5,1),(8,4,1)|#

expand by Row 1

#A=+-1/2[|(5,1),(4,1)|-2|(5,1),(8,1)|+0]#

#A==+-1/2[(5-4)-2(5-8)]#

#A=+-1/2[1+6]#

#A=7/2" "#( taking the positive value)

#:. V=1/3xx6xx7/2#

#=7" units"^3#