Question

The base of a triangular pyramid is a triangle with corners at `(7 ,6 )`, `(4 ,3 )`, and `(1 ,8 )`. If the pyramid has a height of `6`, what is the pyramid's volume?

Answer 1

The volume of the pyramid is 24.

Explanation:

Let `A=(7,6),B=(4,3),C=(1,8)`

When we need to calculate area of an arbitrary triangle with integer coordinates of vertices, it could be useful to draw it and then a rectangle around it. The rectangle should have sides parallel to axes.

![https://www.geogebra.org/geometry]
()

Now we can calculate area of the rectangle and subtract areas of white right triangles. It's much easier than calculating blue triangle's area directly.
`[XYZ]` is the area of shape `XYZ`.

So
`[ABC]=[CDEF]-([BCD]+[ABE]+[ACF])`
`[ABC]=6*5-(1/2*3*5+1/2*3*3+1/2*6*2)`
`[ABC]=30-1/2(15+9+12)`
`[ABC]=30-36/2=30-18=12`

Now we have base area `P=12` and height `h=6` and we can calculate volume of the pytamid by the formula
`V=1/3*P*h=1/3*12*6=24`