Question

The base of a triangular pyramid is a triangle with corners at `(2 ,2 )`, `(3 ,1 )`, and `(7 ,5 )`. If the pyramid has a height of `6`, what is the pyramid's volume?

Answer 1

`8`

Explanation:

To find volume of a triangular pyramid of height `6` and base a triangle with corners at `A(2,2)`, `B(3,1)`, and `C(7,5)` we must find the area of the base triangle.

The sides of triangle can be found as follows.

`AB=sqrt((3-2)^2+(1-2)^2)=sqrt2=1.4142`

`BC=sqrt((7-3)^2+(5-1)^2)=sqrt(16+16)=sqrt32=5.6568`

`CA=sqrt((7-2)^2+(5-2)^2)=sqrt(25+9)=sqrt34=5.831`

Using Heron's formula `s=(1.4142+5.6568+5.831)/2=6.451`

and area of triangle is `sqrt(6.451xx(6.451-1.4142)xx(6.451-5.6568)xx(6.451-5.831)`
i.e. `sqrt(6.451xx5.0368xx0.7942xx0.62)=4` (approx.)

As volume of pyramid `1/3xxheightxxarea of base`, it is `1/3xx4xx6=8`