What is a Kite in Math?

In Euclidean geometry, a Kite is a quadrilateral in which two pairs of adjacent sides are of equal length. The diagonals of the figure intersect each other at right angles. 

• The angles at which the pairs close to the sides meet are equal. 
• It is also known as deltoids or dart.
• There are two types of kites, convex kite and concave kite. 
• Convex kites have their inner angles less than 180°.
• Concave kites have at least one inner angle of more than 180°.
• The sum of the interior angles is 360 degrees.
• Its diagonals are unequal and perpendicular to each other.
• The figure has four non-collinear points.
• Sign boards and diamonds are real-life examples of Kite.

Key Terms: Kite, Right Angles, Convex Kite, Concave Kite, Quadrilateral, Angles, Triangle, Rhombus, Area of Kite, Perimeter of Kite, Diagonal of Kite, Polygon

---

What is a Kite?

Kite is a type of quadrilateral that has two equal angles and two pairs of adjacent equal-length sides. It is a four-sided polygon whose opposite angles are equal and congruent to each other.

• Kite has the greatest ratio of perimeter to diameter with 60°, 75°, and 150° angles.
• It forms reflection symmetry across one of its diagonals.
• Right kites resemble the shape of a cyclic quadrilateral, which consists of a circle that passes through all their vertices.
• It consists of eight polygons that can tile the plane.
• The figure has an axis of symmetry through two opposite angles.

In the quadrilateral ABCD, sides AB = BC and AD = CD. Thus, as the adjacent pairs of sides are equal, therefore ABCD is a specific type of quadrilateral known as a Kite.

• The figure shown below shows a kite:

 Kite 

---

Properties of Kite

[Click Here for Sample Questions]

The properties of the kite are as follows:

• A kite is a form of figure that consists of two congruent triangles having a shared base.
• Both the diagonals intersect each other at right angles.
• The smaller diagonal divides the kite into equal isosceles triangles.
• A kite is a symmetrical figure which consists of equal angles between unequal sides.
• The diagonals are bisected perpendicularly.

---

Proof of Properties of Kite

[Click Here for Sample Questions]

The proof of properties of kite are as follows:

Kite

Statement: The two diagonals of a Kite intersect at right angles.

Proof: From the figure above, in ΔABD and ΔBCD,

• Now, AB=BC
• And, AD=CD
• BD is common.
• Therefore ΔABD ≅ ΔBCD (by SSS rule of congruence)
• Also, in ΔABC and ΔADC
• BC = AB & CD = AD
• Thus, ΔABC is an isosceles triangle.
• ∠BAO = ∠BCO
• Thus, ΔABO ≅ ΔBCO (by ASA rule of congruence)
• ∠AOB = ∠BOC (CPCT)
•  Also, ∠AOB + ∠BOC = 180° (AO & AC are Linear pair)
• ∠AOB = ∠BOC = 90

Statement: One diagonal is the perpendicular bisector of the other diagonal.

Proof: We know,

• ΔABO ≅ ΔBCO
• ∠BOC = ∠AOB = 90°
• AO = OC (Common Parts of Congruent Triangles)
• Thus, we can conclude, AC bisects BD at right angles.

Statement: Angles between unequal sides are equal.

Proof: Since, ΔBAD ≅ ΔBCD

⇒ ∠BAD = ∠BCD (Common Parts of Congruent Triangles)

Also Read:

---

Area of Kite

[Click Here for Sample Questions]

The area of kite is defined as the amount of total space enclosed by a kite in a two-dimensional plane. The area is measured in terms of units², for example, in², cm², m², etc.

• The area of kite is obtained by multiplying half the length of the two diagonals.
• The area is equal to the area of the rhombus.
• To calculate the area of kite, we first need to determine the values of the diagonals.
• We can easily calculate the area of stained glass windows and diamonds.
• If the length of diagonals is given as p and q as shown in the figure given below, the area the of kite can be given as:

Area (A) = (p × q) / 2

---

Perimeter of Kite

[Click Here for Sample Questions]

The perimeter of kite is the distance around its four sides. It is the sum of all the sides of the closed figure. It can also be calculated by adding the sides of each pair.

• The perimeter of kite is twice the sum of the length of AB and BC.
• If we look at the above figure, the four sides are AB, BC, CD & DA. 
• So, the perimeter is the sum of all the sides, i.e., AB+BC+CD+DA.
• Since the adjacent sides are equal, we can say,

Perimeter (P)= 2 x (AB+BC)

---

Diagonals of Kite

[Click Here for Sample Questions]

A kite consists of two diagonals that intersect each other at a right angle. The diagonals of the figure are not equal. The longer diagonal of the figure will divide it symmetrically.

• Longer diagonals will form two congruent triangles by the SSS property of Congruence.
• The shorter diagonal is bisected by a longer diagonal.
• Three sides of the triangle on the left of the longer diagonal are congruent to the sides of the triangle to the right.

---

Things to Remember

• Kite is a quadrilateral where diagonals intersect each other at right angles.
• There are two types of kites, namely, convex kite and concave kite.
• The inner angles of the convex kites are less than 180°.
• Concave kites have at least one inner angle of more than 180°.
• The sum of the interior angles of a kite is equal to 360°. 
• Angles between unequal sides are equal.

Also Read: