Question

The base of a triangular pyramid is a triangle with corners at `(7 ,5 )`, `(6 ,9 )`, and `(3 ,4 )`. If the pyramid has a height of `15`, what is the pyramid's volume?

Answer 1

Area of a triangle given three vertices `(x_1, y_1)` , `(x_2,y_2)`, `(x_3,y_3)` is:

`A = abs((x_1(y_2-y_3) + x_2(y_3-y_1) +x_3(y_1-y_2))/2`

`(x_1,y_1) =(7,5)`
`(x_2,y_2)=(6,9)`
`(x_3,y_3) =(3,4)`

`A=abs((7(9-4)+6(4-5)+3(5-9))/2`

`A=abs((7*-5+6*-1+3*-4)/2`

`A=abs((-35-6-12)/2`

`A=abs(-53/2) =26.5`

Volume of a trianglar pyramid = `V= 1/3Ah` where A is the area of the triangular base and h is the height of the pyramid.

`V=1/3*26.5*15 = 132.5`