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# Kite|Definition & Meaning

## Definition

A **kite** is a quadrilateral **exhibiting** mirror balance when **folded.** Because of this **balance,** a kite possesses two similar **inclinations** and two sets of adjacent **same-length** sides. Alternatively, a Kite is a flat **geometry** having linear sides that **consist** of two pairs of identical-length **adjoining** sides.

**The concept of the geometry of a Kite is shown in Figure 1. **

A **quadrilateral** itself has four corners and four sides, and four angles. The **sum** of all its internal **angles** is 360 degrees. A quadrilateral has sides that have different lengths and different angles. **Squares, rectangles, etc., are particular types of quadrilaterals with some sides and angles equal.**

A Kite is a balanced, closed figure having four linear sides such that there are:

**• Two pairs of sides (in Figure 1, these pairs are AB-CB and AD-CD).****• Each pair of sides consists of two adjacent sides that are equal in length (in Figure 1, |AB| = |CB| and |AD| = |CD|).**

The **angles** are equal where the pairs **meet.**

## Kite Formulas

### Area

The area **represents** the space enclosed by the Kite. The **formula** for area is given by:

**Area (A) = (d1 × d2)/2**

Where the variables** d1** and **d2** represent the length of **diagonals. **Suppose the **diagonals** are 12 m and 16 m in length; the kite area using the above formula, with d1 = 12 cm and d2 = 16 cm, turns out to be:

= (12 $\times$ 16)/2 cm$^2$

= 96 cm$^2$

### Perimeter

**Perimeter** is the total **distance** covered while traveling along the **sides** of the **Kite.** The formula for the **perimeter** for Kite is given by:

**Perimeter (P) = 2(x + y)**

Where x and y are the lengths of the kite’s sides.

For example, suppose you want to find the perimeter of a kite whose side lengths are 10 and 16 cm. From the above formula, substituting x = 10 cm and y = 16 cm gives us:

= 2(10 + 16) cm

= 52 cm

## Sides, Angles, and Diagonals of Kite

### Sides of a kite

A **Kite** has two sets of sides that are **congruent** and the **congruent** pair of Kite sides are not opposing faces.

### Angles of Kite

The angles **subtended** by the neighboring **sides** that are not congruent for the kite are always **congruent.**

### Diagonals of a Kite

A kite has two **diagonals** that are perpendicular to each other:

For kite **WXYZ** as shown in Figure 2, **XW** $\cong$ ZW and XY $\cong$ ZY. Therefore, $\triangle$XYZ and $\triangle$YXZ are isosceles triangles that share a base, XZ. Based on this, we know that the line segment from W and Y to the midpoint of XZ is the height of $\triangle$WXZ and $\triangle$CBD. Therefore, diagonals WY and XZ are perpendicular. Diagonal WY is the perpendicular bisector of diagonal XZ.

## Kite Properties

Kite has two diagonals that cross one another at right angles and is symmetrical around its major diagonal. Angles opposing the major diagonal in a Kite are of the same length. **The Kite can be viewed as a set of congruent triangles having a standard base. The smallest diagonal splits the Kite into two isosceles triangles.**

The **diagonals** of every Kite are at right angles. The **types** of Kites are described in the next sections. When a Kite is of convex type, the sides of the Kite are **tangent** to an inscribed circle. **These include special cases i.e. the right kites, having two opposite right angles; the rhombi, which consist of two diagonal axes of symmetry**; and the squares type of Kites, which are also **special** forms of Kite.

Following are some properties of **Kite** listed point-wise.

- The two
**angles**of the Kite where the unequal**sides**meet are same. - The Kite has two
**diagonals**that cross each other at right**angles.** - The large
**diagonal**of the Kite bisects the other diagonal. **A Kite is symmetrical about its large diagonal.**- The small
**diagonal**divides the Kite into two**isosceles**triangles.

## Types of Kite

**Convex:** The **Kite** is called convex when all of its **interior** angles are less **than** 180$^{\circ}$.

**Concave:** The **Kite** is called concave when at least one of its **interior** angles is greater than 180$^{\circ}$. For example, A dart or an **arrowhead** is a concave Kite.

## Special Cases

### Right Kites

The **right** kites are the Kites that have **two** opposite **right** angles. These **Kites** are cyclic **quadrilaterals,** i.e., there is a **circle** that crosses all their **vertices.** The right **Kite** is shown in Figure 3.

### Equidiagonal Kite

**Compared** with all types of **quadrilaterals,** the form that has the largest **proportion** of its circumference to its dia is known as an **equidiagonal Kite** with certain angles. **Four vertices of this kind of Kite lie at the three corners and another one at the side midpoints of the Reuleaux triangle.**

### A Special Kite

Most of the **time,** there **exist** two sets of **congruent** sides in a **Kite** that are not **congruent,** such a **Kite** is a rhombus that **represents** a special case of **Kite** geometry.

## Duality

**Kites** and isosceles **trapezoids** can seem to be **dual** to each other, **implying** a resemblance between them that **inverts** the dimension of their parts, carrying vertices to sides and sides to vertices. **From any kite, the carved circle is tangent to its 4 sides at the 4 corners of a trapezoid that is isosceles.** This resemblance can also be seen as an **example** of polar exchange, a technique for finding **connected** points with lines and vice versa for a fixed circle.

## Solved Examples of Problems With Kite-shaped Objects

**Example**

Find the **area** of a kite having **diagonal** lengths of 50 cm and 45 cm.

**Solution**

As **stated** above that the area of the **kite** is given as 1/2 ×( d1 × d2), therefore:

**Required** Area = 1/2 × ( 50 × 45)

**Area** =1125 cm$^2$

**Example**

The **area** of a kite-shaped field is 60 cm² and the length of one of its **diagonal** is 6cm. **To cross the field, find the length one has to traverse.**

**Solution**

Given that:

**Area** of a kite = 60 cm$^2$

Length of one **diagonal** = 6 cm

Kite Area = 1/2 × d1 × d2

60 = 1/2 × 6 × d2

d2 = 20 cm

That is, one has to traverse** 20 cm.**

*All images/mathematical drawings were created with GeoGebra.*