Question

The base of a triangular pyramid is a triangle with corners at `(3 ,8 )`, `(4 ,2 )`, and `(9 ,8 )`. If the pyramid has a height of `4`, what is the pyramid's volume?

Answer 1

24

Explanation:

Given coordinates of Triangle ABC

A(3,8)
B(4,2)
C(9,8)

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 = `( 3*(2-8) + 4*(8-8) + 9*(8-2))/2`

Area = `(3*(-6) + 4*0 + 9*6)/2`

=> `(-18+0+54)/2`

=> `36/2` = 18

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=4. So volume = `1/3*18*4` = 24