As you'll recall, the Law of Cosines gives one side of a triangle in terms of the other two sides and the opposite angle. (See Wikipedia here, and why not give them five bucks while you're at it?) So as I sat down with my coffee, I decided there ought to be a Law of Cosines that gives the unknown side of a pentagon in terms of the other four sides and the "opposite angles" (i.e., the three angles of which the unknown side is not a part).

First I treated

**u**,

**v**,

**w**and

**x**as vectors with appropriate orientations; then I computed

*t*-squared as the inner product of

**u**+

**v**+

**w**+

**x**with itself. This led easily to the desired relation, which has some nice rhythms in it pointing to the general case (click to enlarge):

Here is an example problem:

I find

*t*= 1.3826 or so. See how useful the formula is? I don't know how I ever got by in theoretical physics without it.

As an extra stocking stuffer, the simpler Law for quadrilaterals:

Note, these formulas also work for non-simple polygons (closed polygonal chains with self-intersections; see Wikipedia here and you'll have another chance to give them five bucks).

By the way, I donated $10 to Wikipedia myself today. Merry Christmas, Jimmy Wales!

## 3 comments:

P.S.: This was breakfast, not scholarship. For more, here is what I found tonight in a quickie google search: http://www.jstor.org/pss/2688227 and http://www.rowan.edu/colleges/las//departments/math/facultystaff/osler/129%20Law%20of%20Cosines%20Generalized%20Published%20Paper.pdf.

(Oddly, the latter article fails to cite the former.)

Damn, why didn't we put this in the Common Core?

I know! So not college ready.

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