Question

The base of a triangular pyramid is a triangle with corners at `(2 ,5 )`, `(6 ,9 )`, and `(3 ,8 )`. If the pyramid has a height of `15`, what is the pyramid's volume?

Answer 1

`20.001`

Explanation:

To find volume of a triangular pyramid of height `15` and base a triangle with corners at `A(2,5)`, `B(6,9)`, and `C(3,8)` we must find the area of the base triangle.

The sides of triangle can be found as follows.

`AB=sqrt((6-2)^2+(9-5)^2)=sqrt32=5.6568`

`BC=sqrt((6-3)^2+(9-8)^2)=sqrt(9+1)=sqrt10=3.1623`

`CA=sqrt((3-2)^2+(8-5)^2)=sqrt(1+9)=sqrt10=3.1623`

Using Heron's formula `s=(5.6568+3.1623+3.1623)/2=5.9907`

and area of triangle is `sqrt(5.9907xx(5.9907-5.6568)xx(5.9907-3.1623)xx(5.9907-3.1623)`
i.e. `sqrt(5.9907xx0.3339xx2.8284xx2.8284)=sqrt16.0021=4.0002` (approx.)

As volume of pyramid `1/3xxheightxxarea of base`, it is `1/3xx4.0002xx15=20.001`