The base of a triangular pyramid is a triangle with corners at #(5 ,1 )#, #(2 ,3 )#, and #(9 ,4 )#. If the pyramid has a height of #4 #, what is the pyramid's volume?

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Answer 1 · 2016-03-18T12:58:33

Volume of pyramid is #11.336# units

First to find area of base, let us find all the sides of base triangle.

#a=sqrt((2-5)^2+(3-1)^2)=sqrt(9+4)=sqrt13=3.606#

#b=sqrt((9-2)^2+(4-3)^2)=sqrt(49+1)=sqrt50=7.071#

#c=sqrt((9-5)^2+(4-1)^2)=sqrt(16+9)=sqrt25=5#

Now for using Heron's formula, #s=1/2(3.606+7.071+5)=15.677/2=7.8385#

Hence, area of base triangle is

#sqrt(7.8385xx(7.8385-3.606)xx(7.8385-7.071)xx(7.8385-5)#

= #sqrt(7.8385xx4.2325xx0.7675xx2.8385)=8.502#

Hence, Volume of pyramid is #1/3xx8.502xx4=11.336#