The base of a triangular pyramid is a triangle with corners at #(6 ,2 )#, #(5 ,1 )#, and #(7 ,4 )#. If the pyramid has a height of #12 #, what is the pyramid's volume?

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Answer 1 · 2016-07-03T08:51:38

Volume of pyramid is #2.0016# cubic units.

As volume of pyramid one-third of base area multiplied by height, one should first find the area of base triangle.

Here the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula

#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#

and radius of circumscribed circle is #(abc)/(4Delta)#

Hence let us find the sides of triangle formed by #(6,2)#, #(5,1)# and #(7,4)#. This will be surely distance between pair of points, which is

#a=sqrt((5-6)^2+(1-2)^2)=sqrt(1+1)=sqrt2=1.4142#

#b=sqrt((7-5)^2+(4-1)^2)=sqrt(4+9)=sqrt13=3.6056# and

#c=sqrt((7-6)^2+(4-2)^2)=sqrt(1+4)=sqrt5=2.2361#

Hence #s=1/2(1.4142+3.6056+2.2361)=1/2xx7.2559=3.628#

and #Area=sqrt(3.628xx(3.628-1.4142)xx(3.628-3.6056)xx(3.628-2.2361)#

= #sqrt(3.628xx2.2138xx0.0224xx1.3919)=sqrt0.2504=0.5004#

Hence volume of pyramid is #1/3xx0.5004xx12=2.0016#