The base of a triangular pyramid is a triangle with corners at #(2 ,1 )#, #(3 ,6 )#, and #(4 ,8 )#. If the pyramid has a height of #5 #, what is the pyramid's volume?

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Answer 1 · 2017-08-22T18:03:26

#5/2" units"^3#

The volume of a pyramid is given by the formula

#V_P=1/3xx"base area"xx"perpendicular height"#

We are given the height #=5#

so the problem is essentially finding the area of the triangular base.

The most direct way of finding the area of a triangle from its coordinates

#(a,b),(b,c),(d,e)" "#is to find the absolute value of the determinate

#Delta=1/2|(a,b,1) , (b,c,1), (d,e,1)|#

using the coordinates given:

#Delta=1/2|(2,1,1) , (3,6,1), (4,8,1)|#

making the determinant simpler by row operations

#R'_3=R_3-R_1#

#Delta=1/2|(2,1,1) , (3,6,1), (2,7,0)|#

#R'_2=R_2-R_1#

#Delta=1/2|(2,1,1) , (1,5,0), (2,7,0)|#

expanding by#" " C_3#

#Delta=1/2[1|(1,5),(2,7)|-0+0]#

#Delta=1/2(1xx7-5xx2)=1/27-10=-3/2#

So the area of the triangle is#" " 3/2 " units"^2#

#:. V_P=1/3xx3/2xx5#

#V_p=5/2" units"^3#