Question

The base of a triangular pyramid is a triangle with corners at `(6 ,2 )`, `(4 ,5 )`, and `(8 ,7 )`. If the pyramid has a height of `6`, what is the pyramid's volume?

Answer 1

Volume of pyramid is `1/3xx6xx8.0004=16.0008`

Explanation:

Find the distance between corners should give us three sides and using Heron's formula we can then find area of base triangle.

`Delta=sqrt(s(s-a)(s-b)(s-c))`, where `s=1/2(a+b+c)`, where sides of a triangle are `a`, `b` and `c`.

Then area can be multiplied by height and divided by `3`, which will give us volume of pyramid.

The sides of triangle formed by `(6,2)`, `(4,5)` and `(8,7)` are

`a=sqrt((4-6)^2+(5-2)^2)=sqrt(4+9)=sqrt13=3.6056`

`b=sqrt((8-4)^2+(7-5)^2)=sqrt(16+4)=sqrt20=4.4721` and

`c=sqrt((8-6)^2+(7-2)^2)=sqrt(4+25)=sqrt29=5.3852`

Hence `s=1/2(3.6056+4.4721+5.3852)=1/2xx13.4629=6.7315`

and `Delta=sqrt(6.7315xx(6.7315-3.6056)xx(6.7315-4.4721)xx(6.7315-5.3852)`

= `sqrt(6.7315xx3.1259xx2.2594xx1.3463)=sqrt64.0062=8.0004`

Hence volume of pyramid is `1/3xx6xx8.0004=16.0008`