Question

The base of a triangular pyramid is a triangle with corners at `(3 ,8 )`, `(4 ,2 )`, and `(2 ,6 )`. If the pyramid has a height of `7`, what is the pyramid's volume?

Answer 1

volume `= 28/3`

Explanation:

The volume of a pyramid is given by the formula:
`color(white)("XXX")V=1/3 hb` where `h` is the height and `b` is the area of the base.

We are given the height (`h=7`)
so we need to find the base.

Method 1: Use the Pythagorean Theorem to find distance between points then apply Heron's Formula for the Area.
(left as an exercise).

Method 2: Evaluate the Area of the Triangular Base as the Sum and Difference of Trapezoids:

From the relationship between the triangle and the trapezoid segments we see that the area of the base of our pyramid is
`color(white)("XXX")b=7+5-8 =4`

Therefore the Volume of the pyramid is
`color(white)("XXX")V=1/3 xx 7 xx 4 =28/3`