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

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Answer 1 · 2016-09-07T07:20:30

Volume of pyramid is #1/3xx6xx8.0004=16.0008#

Find the distance between corners should give us three sides and using Heron's formula we can then find area of base triangle.

#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#, where sides of a triangle are #a#, #b# and #c#.

Then area can be multiplied by height and divided by #3#, which will give us volume of pyramid.

The sides of triangle formed by #(6,2)#, #(4,5)# and #(8,7)# are

#a=sqrt((4-6)^2+(5-2)^2)=sqrt(4+9)=sqrt13=3.6056#

#b=sqrt((8-4)^2+(7-5)^2)=sqrt(16+4)=sqrt20=4.4721# and

#c=sqrt((8-6)^2+(7-2)^2)=sqrt(4+25)=sqrt29=5.3852#

Hence #s=1/2(3.6056+4.4721+5.3852)=1/2xx13.4629=6.7315#

and #Delta=sqrt(6.7315xx(6.7315-3.6056)xx(6.7315-4.4721)xx(6.7315-5.3852)#

= #sqrt(6.7315xx3.1259xx2.2594xx1.3463)=sqrt64.0062=8.0004#

Hence volume of pyramid is #1/3xx6xx8.0004=16.0008#