Area of Kite

Area of a Kite Shape

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

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

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(900).

  • 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 kite and proof for the area of a 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 intersects 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 = [d1 x d2]/2  

Where, d1 and d2 are the diagonal of the kite

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 diagonal forms the line of symmetry. You calculate the area of the kite by multiplying the two diagonal and divide 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) = [d1 x d2]/2

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

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Now let us see how 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

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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.

Area of Kite = \[\frac{d_{1} d_{2}}{2}\]

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

Quiz Time

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

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2. Calculate the area of this kite:

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3. The length of the shorter diagonal of kite ABCD is 16. Calculate the length of the longer diagonal.


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Facts

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°).

FAQ (Frequently Asked Questions)

1. What is the Alternate Method to Find the Area of the Kite?

Answer: If we know the side lengths and angle between unequal sides, we can use trigonometry to find the area of a kite.Multiply the lengths of two unequal sides by the sine of the angle between them.

 The formula for this is given as:

A = absin(c)


Where a is the length of the short side, b is the length of the long side, and c is the internal angle between those two sides.

Example : Here’s an example of using this formula for a kite with a side a length of 4, a side b length of 7, and an internal angle c value of 100 degrees.

A = absin(c)

A = (4)(7)sin(100°) = (28)(0.9848) = 27.574

2. What is Rhombus?

Answer: Rhombus is a four-sided quadrilateral with all its four sides equal in lengths. Rhombus is also referred to as a slanting square. Opposite sides of a rhombus are parallel. The properties of square and rhombus are similar, the only distinguishing property is that square has all the angles equal to 900, and rhombus does not.

Rhombus is also called an equilateral quadrilateral. Also called a diamond or a lozenge. Every rhombus is a parallelogram but the vice versa is not true.

Some of the important properties of a rhombus are as follows

  • The diagonals of a rhombus bisect each other and are perpendicular(forms right angle).

  • Opposite angles of the rhombus are equal and a diagonal bisects the internal angles.

  • Adjacent sides are equal in lengths.

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