Question

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?

Answer 1

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