Lengths and perimeters (geometric inequalities)

Geometric inequalities involving lengths and perimeters are statements that set bounds (minimum or maximum values) on the lengths of sides or the perimeter of geometric figures, often using properties like the triangle inequality or comparisons between different shapes.

Important properties

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

• Among all polygons with a given number of sides and fixed perimeter, the regular polygon has the largest area.

• Among all closed curves with a given perimeter, the circle encloses the largest area (isoperimetric inequality).

• For rectangles with a fixed perimeter, the square has the largest area.