Library/Geometry/Plane geometry/Geometric inequalities/Quadrilaterals (inequalities)

Quadrilaterals (inequalities)

Overview
Important

Geometric inequalities involving quadrilaterals are statements that compare lengths, angles, areas, or other quantities in a quadrilateral, showing that one is always greater than or less than another. These inequalities often help us understand the limits of what is possible in any quadrilateral, or in special types like parallelograms or rectangles.

Important properties

  • The sum of the lengths of any three sides of a quadrilateral is greater than the length of the fourth side.

  • The area of a convex quadrilateral is maximized when it is cyclic (can be inscribed in a circle) for given side lengths.

  • For any convex quadrilateral with sides a,b,c,da, b, c, d, the area KK satisfies K≤14(a+c)(b+d)K \leq \frac{1}{4}(a + c)(b + d) (Brahmagupta's inequality).

  • The sum of the interior angles of a quadrilateral is always 360∘360^\circ.