Question

The base of a triangular pyramid is a triangle with corners at `(6 ,2 )`, `(3 ,5 )`, and `(4 ,2 )`. If the pyramid has a height of `8`, what is the pyramid's volume?

Answer 1

`V=8" "`cubic units

Explanation:

the volume `V` of a triangular pyramid has the formula

Volume = 1/3 of Area of base * Height of pyramid

Let us compute the area `A` of the base with
`P_1(x_1, y_1)=(6, 2)`
`P_2(x_2, y_2)=(3, 5)`
`P_3(x_3, y_3)=(4, 2)`

`A=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]`

`A=1/2*[x_1*y_2+x_2*y_3+x_3*y_1-x_2*y_1-x_3*y_2-x_1*y_3]`

`A=1/2[(6, 3, 4, 6),(2, 5, 2, 2)]`

`A=1/2*[6*5+3*2+4*2-3*2-4*5-6*2]`

`A=1/[30+6+8-6-20-12]`

`A=1/2(44-38)`

`A=3" "`square units

Let us now compute the volume V of the pyramid

`V=1/3*A*h`

`V=1/3*3*8`

`V=8" "`cubic units

God bless....I hope the explanation is useful.