Kite
A kite is a quadrilateral with two pairs of consecutive equal sides, with perpendicular diagonals where one diagonal bisects the other.
Formula
\text{Area} = (d_1 d_2)/2 \text{ (where } d_1, d_2 \text{ are diagonal lengths)}
Definition
A kite is a four-sided shape with two pairs of sides that are equal, where the equal sides are next to each other rather than across from each other, like a flying kite. It has one line of symmetry, the longer diagonal, along which it can be folded so both halves match, and its diagonals always cross at right angles, with the main diagonal bisecting the other and bisecting the vertex angles at its ends; area $=(d_1 d_2)/2$. A kite is always a tangential polygon, a circle can always be inscribed in it, because its two pairs of adjacent equal sides guarantee the sums of opposite sides are equal.
Example
A diamond shape on a card, an arrowhead, or a flying kite shape are all kites: sides $AB=AD$ (the short pair) and $CB=CD$ (the long pair). For $AB=AD=5$ and $CB=CD=8$, with diagonal $AC$ bisecting diagonal $BD$ at right angles: if $AC=12$ and $BD=6$, the area is $36$, and the angles at $B$ and $D$ are equal. For adjacent pairs $(a,a)$ and $(b,b)$, the main diagonal has length $\sqrt{a^2+b^2-2ab\cos\theta}$, where $\theta$ is half the angle at the wing tips.
Key Insight
A rhombus is a special kite where both pairs of adjacent sides are equal (making all sides equal), and a square is the special kite that is also a rectangle, showing the kite family contains rhombuses, which in turn contain squares. Every kite being tangential (while not every quadrilateral is) places it in a different hierarchy from cyclic quadrilaterals, and the interplay between tangential and cyclic quadrilaterals leads to Poncelet's closure theorem, a profound result in projective geometry about nested conics.