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#