The base of a triangular pyramid is a triangle with corners at #(2 ,2 )#, #(3 ,1 )#, and #(7 ,5 )#. If the pyramid has a height of #6 #, what is the pyramid's volume?

1 Answer
Mar 2, 2016

#8#

Explanation:

To find volume of a triangular pyramid of height #6# and base a triangle with corners at #A(2,2)#, #B(3,1)#, and #C(7,5)# we must find the area of the base triangle.

The sides of triangle can be found as follows.

#AB=sqrt((3-2)^2+(1-2)^2)=sqrt2=1.4142#

#BC=sqrt((7-3)^2+(5-1)^2)=sqrt(16+16)=sqrt32=5.6568#

#CA=sqrt((7-2)^2+(5-2)^2)=sqrt(25+9)=sqrt34=5.831#

Using Heron's formula #s=(1.4142+5.6568+5.831)/2=6.451#

and area of triangle is #sqrt(6.451xx(6.451-1.4142)xx(6.451-5.6568)xx(6.451-5.831)#
i.e. #sqrt(6.451xx5.0368xx0.7942xx0.62)=4# (approx.)

As volume of pyramid #1/3xxheightxxarea of base#, it is #1/3xx4xx6=8#