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

1 Answer
George C.
Jan 5, 2018

#4#

Explanation:

The area of a triangle with vertices #(x_1, y_1)#, #(x_2, y_2)#, #(x_3, y_3)# is given by the formula:

#A = 1/2 abs(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)#

Letting #(x_1, y_1) = (6, 7)#, #(x_2, y_2) = (4, 5)# and #(x_3, y_3) = (8, 7)# we find that the area of the base of the given pyramid is:

#1/2abs(color(blue)(6) * color(blue)(5)+color(blue)(4) * color(blue)(7) + color(blue)(8) * color(blue)(7) - color(blue)(6) * color(blue)(7) - color(blue)(4) * color(blue)(7) - color(blue)(8) * color(blue)(5))#

#=1/2abs(30+28+56-42-28-40) = 2#

Another way of seeing this is by considering the points #(6, 7)# and #(8, 7)# as the base of the triangle, which is of length #2#. Then the apex of the triangle at #(4, 5)# is at height #2# above the base. Then the area of the triangle is:

#1/2 * "base" * "height" = 1/2 * color(blue)(2) * color(blue)(2) = 2#

Then the volume of a pyramid is:

#1/3 * "base" * "height" = 1/3 * color(blue)(2) * color(blue)(6) = 4#

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