The base of a triangular pyramid is a triangle with corners at #(2 ,6 )#, #(5 ,2 )#, and #(8 ,7 )#. If the pyramid has a height of #18 #, what is the pyramid's volume?

1 Answer
Dec 6, 2017

Volume of pyramid # = color (blue)(81.03)# #cm^3#

Explanation:

Volume of a triangular pyramid v = (1/3) * base area * pyramid height.
Pyramid height = 18 cm
Coordinates of the triangular base #(2,6), (5,2), (8,7)#

Area of triangular base = #sqrt(s (s-a) (s-b) (s-c))#
where a, b, c are the three sides of the triangular base and s is the semi perimeter of the base
#s = (a+b+c)/2#

To find triangle sides :

#a = sqrt((5-2)^2 + (2-6)^2) = sqrt(9 +16) = 5#

#b =sqrt ((8-5)^2 + (7-2)^2) = sqrt 34 = 5.831#

#c = sqrt((7-6)^2 + (8-2)^2) = sqrt37 = 6.0828#

#s = (5 + 5.831 + 6.0828) / 2 = 8.4569#

#s-a = 8.4579 - 5 = 3.4579#
#s-b = 8.4579 - 5.831 = 2.6259#
#s-c = 8.4579 - 6.0828 = 2.3751#

Area of base #= sqrt (8.4569*3.4579*2.6259*2.3751)#
Area of triangular base #= 13.505 cm^2#

Volume of pyramid #= (1/3)*13.505*18 = 81.03 cm^3#