The base of a triangular pyramid is a triangle with corners at #(3 ,8 )#, #(4 ,9 )#, and #(5 ,6 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

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Answer 1 · 2018-02-16T03:09:31

Volume of pyramid

#V = (1/3)(A_t * h ) ~~ color(purple)(5.1746)#

Given : Triangle corners A (3,8), B(4,9), C(5,6), height of pyramid h = 7.

To find volume of pyramid.

#Using distance formula let’s find the sides of the base triangle.

#vec(AB) = sqrt((4-3)^2+(9-8)^2) = sqrt2 = 1.4142#

#vec(BC) = sqrt((5-4)^2+(6-9)^2) = sqrt10 = 3.1623#

#vec(CA) = sqrt((5-3)^2 + (6-8)^2) = sqrt13 = 3.6056#

Semi perimeter of triangle base

#s = (a + b + c) / 2 = (1.4142 + 3.1623 + 3.6056)/2 ~~ color(blue)(4.09#

Area of base triangle formula, given three sides is

#A_t = sqrt(s(s-a)(s-b)(s-c)) #

#=> sqrt(4.09(4.09-3.1623)(4.09-3.6056)(1.4142)) ~~ color (green)(2.2177)#

Volume of pyramid

#V = (1/3)(A_t * h )= (1/3) * 2.2177 * 7 ~~ color(purple)(5.1746)#