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

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Answer 1 · 2017-12-28T06:08:49

Volume of pyramid #color(red)v= color(purple)(27)#

Coordinates of three vertices are
A (6,3), B (4,7), C (8,8)
Height of pyramid h = 6

#AB = c = sqrt((4-6)^2 + (7-3)^2) = 4.4721#

#BC = a = sqrt((8-4)^2 + (8-7)^2) = 4.1231#

#CA = b = sqrt ((6-8)^2 + (3-8)^2) = 5.3852#

Semi perimeter #s = (a+b+c)/2 = (4.1231 + 4.4721 + 5.3852) / 2 = 6.9902#

Area of base triangle #= Delta = sqrt (s (s- a) (s - b) (s - c))#

#Delta = #sqrt (6.9902 * (6.9902-4.1231) * (6.9902-4.4721) * (6.990-5.3852)) = #9

Volume of pyramid #v = (1/3) * Delta * h = (1/3) * 9 * 6 = #18