Comments on Area of a (planar) quadrilateral using two opposite sides and four angles

Post

Area of a (planar) quadrilateral using two opposite sides and four angles

−1

I guessed the following area formula for a (planar) quadrilateral using two opposite sides and four angles:(When $a=AB$ and $c=CD$,)

$S = \frac{a^2}{2(\cot A + \cot B)} + \frac{c^2}{2(\cot C + \cot D)}$

This formula can be applied to the quadrilateral satisfying $A+B \ne \pi$ and $C+D \ne \pi$, including convex, concave, and even self-intersecting cases.

I'd like to know the applicable case, inapplicable case and reasons for the validity of those cases. Are there any known papers or more elegant ways to derive this formula?

I guess there are many ways to explain those reasons. It greatly resembles $S = \frac{1}{2}(ab \sin B + cd \sin D)$. It seems that $b$ and $d$ can be determined using $a, c, A, B, C, D$.

geometry euclidean-geometry

posted 4 months ago

CC BY-SA 4.0

4mo ago by clemens‭

$user card$

Ryo TAKA‭

-1 reputation 1 0 -1 3

Raw

Markdown

History

1 comment thread

If you cross-post the same question on multiple sites, you should include links to all other versions... (2 comments)

samcarter‭ wrote 4 months ago

copy link

If you cross-post the same question on multiple sites, you should include links to all other versions of the question in every post, even in the original one. X-posting without explicit links has the risk of wasting both the time of users who might put time and effort into answering a post, which already has an answer elsewhere, as well as the time of users who might have the same problem but can't find the solutions you might have gotten on one of the other sites you x-posted your question

Communities

Comments on Area of a (planar) quadrilateral using two opposite sides and four angles

Area of a (planar) quadrilateral using two opposite sides and four angles

1 comment thread