Question

The base of a triangular pyramid is a triangle with corners at `(1 ,5 )`, `(6 ,2 )`, and `(5 ,9 )`. If the pyramid has a height of `8`, what is the pyramid's volume?

Answer 1

`V=42 2/3` units.

Explanation:

The volume of a pyramid is `V=B*h`, where `B` is the area of the base, and `h` is the height of the pyramid.

The area of the base can be found by subtracting triangles from a rectangle. The graph of the base is shown below.

We can subtract 3 triangles from the the rectangle.

The top left triangle has an area of `1/2*4*4=8`.
The top right triangle has an area of `1/2*1*7=3.5`.
The bottom left triangle has an area of `1/2*3*5=7.5`.

The sum of the areas of these 3 triangles is `8+3.5+7.5=19`. The area of the rectangle is `5*7=35`. So, the area of the base of the pyramid is `35-19=16`.

Plugging this into the formula for the volume, we have `V=1/3*16*8=128/3=42 2/3` units.