Puzzle – area of polygon with known coordinates
2008 May 5
Over at his blog, Dave Marain asked for the fourth coordinate, along with (a,b), (0,0) and (b,a) to give a parallelogram. And then he asked for the area (in at least five ways!)
Later, Dave clarified that he wanted (a+b,a+b). IOW, the origin was to be the second coordinate. And, WLOG assuming b > a > 0, the area of the parallelogram (actually a rhombus) is 
However, without specifying the order of the coordinates, we could have used (a-b, b-a) for the missing coordinate. Can we find the area of this parallelogram?
Can we find the area of any triangle from just its coordinates (0,0), (a,b), (c,d) ? Any arbitrary quadrilateral? I think translating to get one point at the origin is ok, but rotating is not…
Sounds like the shoelace algorithm.
Uh, unless I’m missing something, isn’t this just the three step process, (i) “answer the triangle question”,
(ii) “w.l.o.g., assume one vertex of the polygon is at 0″, and
(iii) “do the completely obvious thing to use the first answer to solve the general problem”,
no?
There’s also Pick’s Theorem.
I hadn’t seen the shoelace algorithm before. It’s a nice little generalisation of the fact that the area of the parallelogram (0,0). (a,b). (c,d). (a+c, b+d) is determinant [a c \\ b d] (so triangles follow, and all other polygons follow that).
Pick’s Theorem is a truly splendid thing.
1. find for the area by green’s theorem or determinants
2. use pick’s theorem
3. to find for the vertex, [...]