Polygons (inequalities)

Geometric inequalities for polygons are statements that relate the lengths of sides, the measures of angles, or other properties (like area or perimeter) using inequalities. The most famous is the triangle inequality, but there are also inequalities for quadrilaterals and other polygons.

Important properties

• Triangle inequality: In any triangle with sides $a$, $b$, $c$, we have $a + b > c$, $a + c > b$, and $b + c > a$.

• For any convex polygon, the sum of the lengths of any set of sides is greater than the length of the remaining side.

• The sum of the interior angles of an $n$-gon is $(n-2) \times 180^\circ$; each angle must be less than $180^\circ$ in a convex polygon.

• In a convex quadrilateral, the sum of any three sides is greater than the fourth side.