This innocent problem can be solved without vector operations or trigonometry. (But that doesn't mean that one or both of those tools isn't useful.) The followup question requires trigonometry and either vector operations or a good understanding of coordinate systems.
Area of a Polygon
I was recently asked for an equation giving the area of an arbitrary quadrilateral. I immediately generalized this to an arbitrary polygon and started on the problem. This led to some very elegant and useful results.
Given: a polygon defined by points (x1, y1) through (xn, yn). The polygon is composed of lines between adjacent vertices, and (x1, y1) is adjacent to (xn, yn). What is the area of the polygon in terms of the coordinates of the vertices?
I found a very elegant solution without much work. It turns out that I was a bit lucky in the approach that I chose, but there are a few ways to get there. Just one hint: Do not start with the semiperimeter formula. If that's not a dead end, it's at least too much work.
The follow-up question is a bit more difficult, and the result is not as elegant. Given: an arbitrary polygon on the unit sphere defined by points (θ1, φ1) through (θn, φn) in spherical coordinates. Each side of the polygon is the segment of a great circle. For example, the line from (π/4, 0) to (π/4, π/2) is not along the path (π/4, φ). Like jet aircraft flight plans, we don't travel directly east to get to a point directly east. The side of the polygon is the shortest great circle route (i.e., the path with distance less than π).
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