The base of a triangular pyramid is a triangle with corners at $(3 ,4 )$ , $(2 ,9 )$ , and $(5 ,7 )$ . If the pyramid has a height of $7$ , what is the pyramid's volume?

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Answer 1 · 2016-08-06T19:51:34

$91/6$

Given

$p_1 = {3,4,0}$
$p_2 = {2,9,0}$
$p_3 = {5,7,0}$

and deffining

$vec u = p_1-p_2$ and
$vec v = p_3-p_2$

The area spanned by two vectors $vec u, vec v$ with the same origin, has a vectorial formula.

$A = 1/2 norm(vec u xx vec v) = 13/2$

The pyramid volume is given by

$V = 1/3 h A = 1/3 cdot 7 cdot 13/2 = 91/6$

The base of a triangular pyramid is a triangle with corners at (3 ,4 )(3 ,4 ), (2 ,9 )(2 ,9 ), and (5 ,7 )(5 ,7 ). If the pyramid has a height of 7 7 , what is the pyramid's volume?