The base of a triangular pyramid is a triangle with corners at #(2 ,4 )#, #(3 ,2 )#, and #(5 ,5 )#. If the pyramid has a height of #5 #, what is the pyramid's volume?

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Answer 1 · 2018-05-17T06:46:57

Volume of triangular base pyramid #color(violet)(V = 466.47# cub. units

#A (2,4), B (3,2), C (5,5)#

Using distance formula,

#bar (AB) = c = sqrt((3-2)^2 + (2-4)^2) = sqrt5 = 2.235#

#bar (BC) = a = sqrt((5-2)^2 + (5-4)^2) = sqrt 10 = 3.162#

#bar (AC) = b = sqrt((3-5)^2 + (2-5)^2) = sqrt 13= 3.606#

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

Semi perimeter #s (a -+ b+ c) / 2 = (2.235 + 3.162 + 3.606)/2 = 18#

#A_b = sqrt(18* 15.765 * 16.838 * 16.394) = 279.88#

Formula for volume of pyramid #V = (1/3) * A_b * h#

# color(violet)(V = (1/3) * 279.88 * 5 = 466.47# cub. units.