The base of a triangular pyramid is a triangle with corners at #(7 ,6 )#, #(4 ,1 )#, and #(3 ,2 )#. If the pyramid has a height of #6 #, what is the pyramid's volume?

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Answer 1 · 2018-01-27T09:06:03

#color(brown)(V = 32# cubic units

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

a, b c are the lengths of three sides and s the semi perimeter of the triangle #s = (a + b + c) /2#

To find the three sides by applying the distance formula,

#d = sqrt((x2-x1)^2 + (y2-y1)^2)#

#a = sqrt((3-4)^2 + (2-1)^2) = sqrt2 ~~ color(red)(1.4142#

#b = sqrt((3-7)^2 + (2-6)^2) = sqrt32 ~~ color(red)(5.6568#

#c = sqrt((4-7)^2 + (1-6)^2) = sqrt34 ~~ color(red)(5.831#

Semi perimeter of triangle

#s = (1.4142 + 5.6568 + 5.831) / 2 = color(red)(6.451)#

Area of triangle base

#B = sqrt(6.451 (6.451 - 1.4142) ( 6.451 - 5.6568) (6.451 - 5.831))#

#color(green)(B ~~ 16)# sq units

Volume of triangle based pyramid #V = (1/3) B h# where h is the height of the pyramid.

#color(brown)(V = (1/3) 16 * 6 = 32)# cubic units