Area of Kite: Formula, Proof & Solved Problems

We have studied that Rhombus is a four-sided quadrilateral with all its four sides equal in length. Rhombus is a kite with all its four sides congruent.

A kite is a special quadrilateral with two pairs of equal adjacent sides. 

The space encircled by a kite is known as the kite area. A kite is a quadrilateral with two pairs of equal sides on each side. A kite's elements are its four angles, four sides, and two diagonals. We shall concentrate on the area of a kite and its formula in this post.

The area of a kite in a two-dimensional plane can be described as the amount of space enclosed or surrounded by the kite. A kite, like a square or a rhombus, does not have equal sides on all four sides. A kite's area is always represented in terms of units^2, such as in^2, cm^2, m^2, and so on.

Properties of a Kite:

• Opposite Angles between unequal sides are equal.

• A kite has two pairs of congruent triangles with a common base.

• Diagonals of a kite intersect each other at right angles(90°).

• The diagonals bisect each other perpendicularly.

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In this article let us study how to find the area of a kite shape , formula for the area of the kite, and proof for the area of the kite.

Mathematically speaking, in the case of building your kite, the area of the kite is the size of the fabric needed to build your kite. And the pieces of wood in our kite diagonals. Diagonals are the two lines that intersect perpendicularly to one another. To find the area of a kite we have, formula for the area of the kite that only requires lengths of the diagonals of the kite.

Area of a Kite = \[\frac{\left [ d_{1} \times d_{2}\right ]}{2}\]

Where, d1 and d2 are the diagonal of the kite.

Formula of Area of a Kite

To find the area of a kite we must know the values of its diagonal. The diagonals of a kite bisects perpendicularly to each other. One of its diagonals forms the line of symmetry. You calculate the area of the kite by multiplying the two diagonals and dividing it by 2.

Area of kite is given as half the product of its diagonal . and it is expressed as 

Area of a Kite = ½ (d₁ x d₂) = \[\frac{\left [ d_{1} \times d_{2}\right ]}{2}\]

Where d₁ and d₂ are the two diagonals of the kite.

Now let us see the derivation of the kite formula.

Proof for Area of a Kite

To find the area of a kite, we will use the below figure of a kite with diagonals d1 and d2 and a line of symmetry d₁. As d₁ is the line of symmetry it divides the kite into two equal triangles, ABC and ADC

Step 1:

We have,

Area of Kite  =  area of ABC + area of ADC

Step 2:

Area of triangle ABC = ½ (base × height)

base = d₁

height = OB 

Area of triangle ABC  = ½ x d₁ x OB……………..(1)

Step 3:

Area of triangle ADC = ½ (base × height)

base = d₁

height = OD 

Area of triangle ADC  = ½ x d₁ x OD……………(2)

Step 4:

Adding 1 and 2

Area of Kite   =  area of ABC + area of ADC

=  ½ x d₁ x OB +  ½ x d₁ x OD

= ½ d₁( OB + OD)

but , OB + OD = BD = d₂….(given)

Therefore Area of Kite = ½ d₁ x d₂

Hence proved

Once you know the length of the diagonals, you can just multiply them and divide the result by 2.

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Solved Examples

1. Find the area of a kite with diagonals that are 6 inches and 18 inches long.

Solution:

Area of a kite = d₁d₂/ 2 

=  (6 × 18) / 2 

= 108 / 2 

= 54 square inches.

2. When the diagonals of a kite meet, they make 4 segments with lengths 6 meters, 4 meters, 5 meters, and 4 meters. What is the area of the kite?

Solution:

The segments with lengths 4 meters and 4 meters must represent the segment that was bisected into 2 equal pieces or d₂

Therefore 

d₂ = 4 + 4 = 8 meters

The segments with lengths 6 meters and 5 meters must represent d1 then

d₁ = 6 meters + 5 meters = 11

Area of a kite = ½ ( d1 x d2)

= (8 × 11) / 2 

= 88 / 2 

= 44 square meters

3. At a park, four friends are flying kites of the same size. The diagonals of each kite are 12 inches and 15 inches. Find the total area of four kites.

Solution:

Lengths of diagonals are:

d₁ = 12in

d₂ = 15in

The area of each kite is:

A =  ½ × d₁ × d₂

    = ½ × 12 × 15

    = 90 in²

Because each kite has the same size, the overall area of all four kites is equal to 4 × 90 = 360in²

The area of the four kites is therefore 360in²

4. Sam wants to offer his buddy a kite-shaped chocolate box. he wants to cover the top of the box with a photo of himself and his friend. Calculate the area of the top of the box if the lid's diagonals are 9 in and 12 in.

Solution:

d₁ = 9in

d₂ = 12in

Because the box is kite-shaped, the area of the top of the box is equal to:

A =  ½ × d₁ × d₂

    = ½ × 9 × 12

Therefore, the area of the top of the box is 54in2

Quiz Time

1. Find the area of a kite with diagonals of 12 inches and 18 inches. 

Solution: 

Area of a kite =d₁d₂/ 2 

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= (12 x 18) / 2

= (216) / 2

= 108 square inches.

2. Calculate the area of this kite: 

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Solution:

Area of Kite =  ½ × d₁ × d₂

Putting the values, we get, 

= ½ х 4 ×ٖ 5

= ½ х 20 = 10 m²

3. The Area of a kite is 126 cm² and one of its diagonals is 21cm long. Calculate the length of the other diagonal.

Solution:

As given in the question,

Area of a kite =126 cm²

Length of one diagonal = 21 cm

Area of Kite =  ½ × d₁ × d₂

126 = ½ x 21 × d₂

  d₂ = 12 cm

Facts

The formula for kite also works for finding the area of a rhombus, and the area of a square since a rhombus is a particular kind of kite (one where all four sides are congruent) and a square is a particular kind of rhombus (where all angles are 90°).

Key Notes

• The perimeter of a kite is: 2(Side₁ + side₂)

• Because a kite is a cyclic quadrilateral, it satisfies all of the cyclic quadrilateral's qualities.