Area of Kite – Derivation, Formulas, Examples

What is a kite?

A kite is an irregular quadrilateral with 2 equal pairs of sides adjacent to each other.Both the diagonals of the kite intersect each other at right angles.

Here are some properties of a kite

• In the figure above, JK = KL and JM = LM, which means that adjacent sides of a kite are equal in length.
• Diagonals of a kite make an angle of 90 degrees at the point of intersection.
• Angles between unequal sides are equal. Here, ∠J = ∠K
• The longer diagonal divides the kite into one pair of congruent triangle.

What is the area of a kite formed by two perpendicular diagonals?

Half the product of its diagonals gives the formula for the area of a kite. For example, if the diagonals of a kite are D1 and D2, then:

Area of kite = ½ D1 x D2

Let us find the area of the kite PQRS.

Here the diagonals are PR and QS

Let diagonal PR be ‘a’ and diagonal QS be ‘b.’

We know the diagonals of a kite bisect each other at right angles.

So, in the figure, diagonal PR bisects diagonal QS.

OQ = OS = OS/2 =  b/2

Area of the kite = sum of area of triangle PQR and PSR

Area of Triangle = ½ b×h

baseis‘a’ and height is OQ = OS = OS/2 =  B/2

Area of triangle PQR = ½  x a x b/2

Area of triangle PSR= ½  x a x b/2

So, area of kite = ½  x a x b/2+ ½  x a x b/2

= ab/4  + ab/4

= 2ab/4 = 1/2ab

So, the formula for the area of the kiteis ½  x diagonal1 x diagonal2 = half the product of diagonals

An alternate method to find the area of a kite

Consider that you do not know the measure of the kite’s diagonals; how will you solve the area of kite questions without the diagonals if you do not know any alternate methods?

Here’s the catch, you can use trigonometry to find the area of the kite formula with sides and their included angle. You can use the formula,

Area of kite = ab sinC

where

a and b denote the measure of unequal sides

c is the internal angle between them

and sin is the sine function in trigonometry

Examples

Example 1

How can you prove the area of the kite is equal to the area of the rhombus?

Solution

We explain to you how tocalculate the area of a rhombus by drawing two diagonals d1 and d2. These two diagonals are perpendicular to each other and bisect each other. So, one diagonal splits the rhombus into two equal triangles. Hence, the area of a rhombus is equal to the sum of the area of these two triangles.

So, area of triangle ADC = area of triangle ABC = ½  x d1 x d2/2

Area of rhombus = area of ADC + area of triangle ABC

Area of rhombus = ½ x d1 x d2/2 + ½  x d1 x d2/2

= ½ x d1 x d2

Considering this, the area of the kite and rhombus is the same.

Example 2

Calculate the area of a kite with diagonals 24cm and 10cm

Solution

Area of kite =  ½ x d1 x d2

=½  x 24 x 10 cm²

= 120 cm²

Therefore, the area of kite = 120 cm²