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

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Answer 1 · 2016-10-03T15:18:10

#V = 10/3 units^3#

Let #(A_x, A_y) = (5, 8)#
Let #(B_x, B_y) = (3, 4)#
Let #(C_x, C_y) = (4, 8)#

According to Area of a triangle given 3 points the area, #Delta#, of the base is:

#Delta = |(A_x(B_y - C_y) + B_x(C_y - A_y) + C_x(A_y - B_y))/2|#

#Delta = |(5(4 - 8) + 3(8 - 8) + 4(8 - 4))/2|#

#Delta = |(5(-4) + 3(0) + 4(4))/2|#

#Delta = |(-20 + 16)/2|#

#Delta = |(-4)/2|#

#Delta = 2#

The volume of the pyramid

#V = 1/3 Deltah# where h is the height

#V = 1/3 (2)(5)#

#V = 10/3 units^3#