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Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12. [ 2 ] Characterizations
Alternatively, the area can be calculated by dividing the kite into two congruent triangles and applying the SAS formula for their area. If a {\displaystyle a} and b {\displaystyle b} are the lengths of two sides of the kite, and θ {\displaystyle \theta } is the angle between, then the area is [ 27 ] A = a b ⋅ sin θ . {\displaystyle ...
where K is the area of a convex quadrilateral with perimeter L. Equality holds if and only if the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter. The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral. [43]
giving the basic form of Brahmagupta's formula. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths. A related formula, which was proved by Coolidge, also gives the area of a general convex
A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common. [17]: p. 84
The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals. [ 1 ] A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a parallelogram ).
A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following: [6] [7] a parallelogram in which a diagonal bisects an interior angle; a parallelogram in which at least two consecutive sides are equal in length; a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
Kites are examples of ex-tangential quadrilaterals. Parallelograms (which include squares, rhombi, and rectangles) can be considered ex-tangential quadrilaterals with infinite exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides (since they are parallel). [4]