The base of a triangular pyramid is a triangle with corners at #(1 ,2 )#, #(5 ,5 )#, and #(2 ,7 )#. If the pyramid has a height of #9 #, what is the pyramid's volume?

1 Answer
Jun 26, 2016

25.5

Explanation:

Given coordinates of Triangle ABC

A(1,2)
B(5,5)
C(2,7)

Given the coordinates of the three vertices of any triangle, the area of the triangle is given by:
area =
#(Ax*( By − Cy ) + Bx*( Cy − Ay ) + Cx *( Ay − By ) )/2#

where Ax and Ay are the x and y coordinates of the point A etc..

In our case, area = #( 1*(5-7) + 5*(7-2) + 2*(2-5))/2#

Area = # (1*(-2) + 5*5 + 2*(-3))/2#

=> #(-2+25-6)/2#

=> #17/2# = 8.5

The volume of a triangular pyramid is V = 1/3AH where A = area of the triangle base, and H = height of the pyramid or the distance from the pyramid's base to the apex.

In our case, H=9. A=8.5 So volume = #1/3*8.5*9# = 25.5