If I had to pick a favorite formula, it would where stands for the area of a polygon with vertex coordinates in order . The absolute value sign can be omitted if the inside of the polygon is kept to the left as the vertices are traversed.
I first encountered this formula in David Cohen’s Precalculus, 3rd edition. The problem sets in this text are outstanding. (Thanks to William Bucher for first suggesting it.) They are diverse and explicitly sorted into A, B, and C levels. The C level problems are particularly interesting. The formula applied to a triangle is given as part of an exercise on page 54 exercise 14(b) and page 56 exercise 55.
One year my Precalculus students did a project where, given the coordinates around a polygonal lake, they found the area in at least three ways. I expected them to divide the polygon into triangles and use various methods of calculating the areas. Previously I had given them one of Cohen’s problems so I expected them to also to use my favorite formula and find that it was the easiest method. Two of my students working as a group found the area in seven different ways. Their project was complete with an extensive notebook and a fifteen foot long butcher paper illustrated poster.
My favorite application of the ideas in the formula is Green’s theorem in the plane. As can be seen here, the effects along the internal sides of the triangles cancel out. This is really cool. I like the idea that knowing what happens on the edges tells you what is going on inside. Green’s theorem can also be used to get the area of a plane figure as described here. Note the similarity to the formula for the area of a polygon.
I recently (last weekend) used the polygonal formula in a Python program I wrote to find the area of expansion of a granitic intrusion working with Dr. Bill Hirt of the College of the Siskiyous.