Question

The base of a triangular pyramid is a triangle with corners at `(3 ,4 )`, `(6 ,2 )`, and `(5 ,7 )`. If the pyramid has a height of `7`, what is the pyramid's volume?

Answer 1

The volume is `V = 91/6color(white).units^3`

Explanation:

The volume, V, of a triangular pyramid is:

`V =(1/3)Ah`

where A is the area of the base and h is the height.

We are given `h = 7`

Therefore, we need a formula for the area of a triangle given the coordinates of its vertices :

`Area = |(A_x(B_y − C_y) + B_x(C_y − A_y)+ C_x(A_y − B_y))/2|`

`Area = |(3(2 − 7) + 6(7 − 4)+ 5(4 − 2))/2|`

`Area = |(3(2 − 7) + 6(7 − 4)+ 5(4 − 2))/2|`

`Area = 13/2`

`V =(1/3)(13/2)7`

`V =(1/3)(13/2)7`

`V = 91/6color(white).units^3`