The base of a triangular pyramid is a triangle with corners at #(2 ,6 )#, #(5 ,4 )#, and #(7 ,5 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

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Answer 1 · 2018-06-09T02:04:00

#color(green)("Volume of Pyramid " V_p = (1/3) * A_b * h = 8.29 " cubic units"#

#color(violet)("Volume of Pyramid " V_p = (1/3) * A_b * h#

#Area of base triangle " A_b = sqrt(s (s-a) (s-b) (s-c)), " using Heron's formula#

#A(2,6), B(5,4), C(7,5), h = 7#

#a = sqrt((5-7)^2 + (4-5)^2) = 2.24#

#b = sqrt((7-2)^2 + (5-6)^2) = 5.1#

#c = sqrt((2-5)^2 + (6-4)^2) = 3.61#

#"Semi-perimeter " s = (a + b + c) / 2 #

#s = (2.24 + 5.1 + 3.61) / 2 ~~ 5.48#

#A_b = sqrt(5.48 * (5.48 - 2.24) * (5.48 - 5.1) * (5.48 - 3.61)) = 3.55#

#color(green)("Volume of Pyramid " V_p = (1/3) * A_b * h = (1/3) * 3.55* 7 = 8.29 " cubic units"#