The base of a triangular pyramid is a triangle with corners at #(8 ,5 )#, #(7 ,2 )#, and #(4 ,6 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

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Answer 1 · 2018-07-30T03:08:28

#color(green)(V = 1/3 A_b h = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"#

Given : #A (8,5), B (7,2), C (4,6)#, h = 7#

Using distance formula we can calculate the lengths of sides a, b, c.

#d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)#

#a = sqrt((7-4)^2+(2-6)^2) = 5#

#b = sqrt((4-8)^2+(6-5)^2) = sqrt 17 #

#c = sqrt((8-7)^2+(5-2)^2) = sqrt 10#

Semi perimeter of base triangle #s = (a+b+c)/2#

#s = (5 + sqrt 17 + sqrt 10)/2 = 12.2854/2 ~~ 6.2427#

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

#A_t = sqrt(6.2427 * (6.2427-5)*(6.2427- sqrt 17)*(6.2427-sqrt 10))#

#A_t ~~ 7.1171

Volume of pyramid #V = (1/3) * A_t * h#

#color(green)(V = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"#