Kite (geometry) facts for kids

Quick facts for kids Kite
A kite, showing its pairs of equal-length sides and its inscribed circle.
Type	Quadrilateral
Edges and vertices	4
Symmetry group	D1 (*)
Area	1/2 × product of diagonals
Dual polygon	Isosceles trapezoid

A kite is a special type of quadrilateral, which is a shape with four straight sides. You might recognize this shape from the flying toys called kites!

What makes a kite shape special? It has reflection symmetry across one of its main lines, called a diagonal. This means if you fold the kite along this diagonal, both halves match perfectly. Because of this symmetry, a kite always has two pairs of sides that are equal in length, and these equal sides are always next to each other. It also has two equal angles.

Kites are interesting shapes in Euclidean geometry. For example, their diagonals (lines connecting opposite corners) always cross each other at a perfect right angle (90 degrees). Also, if a kite is convex (meaning it doesn't have any "dents" pushed inwards), you can always draw a circle inside it that touches all four of its sides.

What is a Kite?

Kites can be convex (all angles pointing outwards) or concave (with one angle pointing inwards, like a dart).

A kite is a four-sided shape where its sides can be grouped into two pairs of equal-length sides. These equal sides must be next to each other. Think of it like this: two short sides meet at one corner, and two long sides meet at another corner.

Here are some ways to recognize a kite:

• It has two pairs of adjacent sides that are the same length.
• One of its diagonals acts as a line of symmetry. This diagonal cuts the kite into two identical mirror-image triangles.
• This special diagonal also cuts the angles at its ends exactly in half.
• The two diagonals always cross each other at a 90-degree angle.

The name "kite" comes from the flying toys we see in the sky, which often have this shape. These toys were named after a type of bird that hovers in the air.

Different Kinds of Kites

Some shapes are special types of kites. For example, a rhombus (a shape with four equal sides) is a kite. A square is also a kite, because all its sides are equal, and they are adjacent.

In this article, we consider rhombi and squares to be special kinds of kites. This helps us understand their properties more easily.

Kites are different from parallelograms. Parallelograms also have two pairs of equal-length sides, but those sides are opposite each other, not next to each other.

Special Kite Shapes

A right kite has two opposite right angles.

An equidiagonal kite.

A kite used in the "Lute of Pythagoras" fractal pattern.

Right Kites

A right kite is a kite that has two opposite right angles (90-degree angles). These special kites can also fit perfectly inside a circle, meaning all their corners touch the circle. Shapes that can be drawn inside a circle like this are called cyclic quadrilaterals.

Kites in Patterns

Some kites are used to create amazing patterns! For example, two specific kite shapes (one convex and one concave, like a dart) are the main pieces for a famous pattern called the Penrose tiling. This tiling is special because it can cover a flat surface without ever repeating itself in a regular way. It's like a puzzle that never shows the same part twice!

One interesting kite has three angles of 108 degrees and one angle of 36 degrees. This kite is part of a complex pattern called the "Lute of Pythagoras," which is a type of fractal (a pattern that repeats itself at different scales).

Another kite, with angles of 60, 90, 120, and 90 degrees, can tile a flat surface by simply reflecting it over its edges. This creates a beautiful pattern known as the deltoidal trihexagonal tiling.

How Kites Work: Properties

Diagonals, Angles, and Area

The two diagonals of a kite always cross each other at a right angle (90 degrees). One of these diagonals is also the kite's line of symmetry. This diagonal cuts the other diagonal exactly in half. It also cuts the two angles it passes through into two equal parts. The other two angles of the kite (the ones not on the symmetry diagonal) are always equal to each other.

To find the area of a kite, you can use a simple formula. If you multiply the lengths of its two diagonals (let's call them p and q) and then divide by two, you get the area: You can also find the area if you know the lengths of two different sides (say, a and b) and the angle (let's call it ) between them. The formula is:

Circles Inside and Outside Kites

Circles that touch the sides of a convex kite (top), a non-convex kite (middle), and an antiparallelogram (bottom).

Every convex kite (one without inward dents) can have a circle drawn inside it that touches all four of its sides. This is called an inscribed circle.

If a convex kite is not a rhombus, it can also have another circle outside it that touches the extensions of its four sides.

For concave kites (like a dart shape), there are also circles that touch its sides or the lines that extend its sides.

A four-sided shape that has an inscribed circle is called a tangential quadrilateral. A tangential quadrilateral is a kite if its diagonals cross at a right angle. It's also a kite if the center of its inscribed circle lies on one of its diagonals (the symmetry diagonal).

Kites and Their "Partner" Shapes

Kites have a special relationship with isosceles trapezoids. They are considered "dual" shapes. This means that if you take a kite and find the points where its inscribed circle touches its sides, those four points will form an isosceles trapezoid. And if you do the opposite for an isosceles trapezoid, you'll get a kite!

Here's a quick comparison of their features:

Kites in Tilings and 3D Shapes

How the kite and dart Penrose tiling can be built step-by-step.

A fractal pattern made from Penrose kites.

Kites are great for making patterns! All kites can be used to tile a flat surface (like a floor) by repeating them.

As mentioned before, special kites and dart shapes are the building blocks for the famous Penrose tiling. This tiling creates beautiful, non-repeating patterns. You can even make fractal patterns, like rosettes, using these kites.

Kites also form the faces of many interesting 3D shapes called polyhedra.

• The deltoidal icositetrahedron and deltoidal hexecontahedron are examples of polyhedra whose faces are all congruent (identical) kite shapes.
• The trapezohedron is another family of polyhedra with kite-shaped faces. A common example is the ten-sided dice, which is a type of pentagonal trapezohedron.

Here are some examples of how kites are used to tile surfaces or form faces of 3D shapes:

3D Shapes (Polyhedra)			Flat Tilings
Rhombic dodecahedron	Deltoidal icositetrahedron	Deltoidal hexecontahedron	Deltoidal trihexagonal tiling
3D Shapes	Flat Tilings	Hyperbolic Tilings
Deltoidal icositetrahedron	Square tiling	Deltoidal tetrapentagonal tiling	Deltoidal tetrahexagonal tiling
3D Shapes	Hyperbolic Tilings
Deltoidal hexecontahedron	Deltoidal tetrapentagonal tiling	Rhombic pentapentagonal tiling	Deltoidal pentahexagonal tiling
Flat Tilings	Hyperbolic Tilings
Deltoidal trihexagonal tiling	Deltoidal tetrahexagonal tiling	Deltoidal pentahexagonal tiling	Rhombic hexahexagonal tiling

Ten-sided dice often have the shape of a pentagonal trapezohedron, which has kite-shaped faces.

See also

In Spanish: Deltoide para niños