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)

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

$=7" units"^3$

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