The base of a triangular pyramid is a triangle with corners at $(7 ,8 )$ , $(5 ,3 )$ , and $(8 ,4 )$ . If the pyramid has a height of $6$ , what is the pyramid's volume?

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Answer 1 · 2017-10-17T15:28:48

$13" units"^3$

the volume of a pyramid

$V=1/3xx"base area"xx "perpendicular height"$

in the question we have the height and the coordinates of the triangle's vertices.

To find the area of a triangle with coordinates

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

we evaluate the determinant

$A_(Delta)=1/2|(1,1,1),(x_1,x_2,x_3),(y_1,y_2,y_3)|$

for this question:

$A_(Delta)=1/2|(1,1,1),(7,5,8),(8,3,4)|$

expanding by $R_1$

$A_(Delta)=1/2[|(5,8),(3,4)|-|(7,8),(8,4)|+|(7,5),(8,3)|]$

$A_(Delta)=1/2[-4+36-19]$

$A_(Delta)=1/2xx13=13/2$

the volume of teh pyramid is therefore

$V=1/cancel(3)xx13/cancel(2)xxcancel(6)$

$13" units"^3$

The base of a triangular pyramid is a triangle with corners at (7 ,8 )(7 ,8 ), (5 ,3 )(5 ,3 ), and (8 ,4 )(8 ,4 ). If the pyramid has a height of 6 6 , what is the pyramid's volume?