Kite
GeometryA 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 (not across from each other). It looks like a flying kite.
Example
A diamond shape on a card, an arrowhead, or a flying kite shape are all kites. Two pairs of adjacent sides are equal: sides $AB = AD$ (the "short pair") and sides $CB = CD$ (the "long pair").
Key Insight
A kite has one line of symmetry - the longer diagonal. You can fold the kite along this diagonal and both halves match. The diagonals of a kite always cross at right angles.
Definition
A kite is a quadrilateral with two pairs of consecutive (adjacent) congruent sides. One pair of opposite angles is congruent (the angles between the unequal sides). The diagonals are perpendicular; the main diagonal (symmetry axis) bisects the other diagonal and bisects the vertex angles at its ends.
Example
Kite with $AB = AD = 5$ and $CB = CD = 8$. The diagonal $AC$ bisects diagonal $BD$ at right angles. Area $= (AC \times BD)/2$. If $AC = 12$ and $BD = 6$: area $= 36$. The angles at $B$ and $D$ are equal.
Key Insight
A rhombus is a special kite where both pairs of adjacent sides are equal (all sides equal). A square is the special kite that is also a rectangle. This shows that the kite family contains rhombuses, which in turn contain squares.
Definition
A kite has a reflection symmetry across one diagonal (the "main" diagonal), which bisects the other diagonal perpendicularly. Its area is $(d_1 d_2)/2$ because the perpendicular diagonals create four right triangles whose areas sum to this value. A kite is a tangential polygon: a circle can be inscribed in it (the incircle is tangent to all four sides) iff the sum of opposite sides are equal - and for a kite, $AB + CD = AD + BC$ always holds (since $AB = AD$ and $BC = CD$ by definition... actually $AB + CD = AB + CD$), so a kite is always tangential.
Example
For kite with 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. The perpendicular diagonal length $= 2\sqrt{a^2 - d_1^2/4}$.
Key Insight
Every kite is a tangential polygon (an inscribed circle exists), while not every quadrilateral is. This makes kites part of a different hierarchy from cyclic quadrilaterals. The interplay between tangential and cyclic quadrilaterals leads to Poncelet's closure theorem, a profound result in projective geometry about nested conics.