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 $" " 3/2 " units"^2$

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

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

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