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 $1/3xx0.5004xx12=2.0016$

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