Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?

share|cite|improve this question
I'm not sure why this was edited just now - it is clear from JoR's answer that if the longest edge is equal to the sum of the others, the area is zero. On the other hand, if the area is zero, it does not imply that this condition holds, for example all edges could be the same length. – Carl Apr 30 '14 at 6:38
I interpreted the question differently in my answer, but I see now that other interpretations are possible. – Joseph O'Rourke Apr 30 '14 at 15:39

1 Answer 1

up vote 6 down vote accepted

A chain of edges can close iff the longest edge is not longer than the sum of the lengths of all the other edges. This is Theorem 8.6.3 (p.326) in Computational Geometry in C and Theorem 5.1.2 (p.61) in Geometric Folding Algorithms. You can easily see the necessity of this condition: If one edge $e$ is too long, the others all together cannot reach from end-to-end of $e$.

This result has nothing to do with $6$; it holds for $n \ge 3$ edges.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.