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 $1/3xx8.502xx4=11.336$

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