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Diophantine Kites: Rational Diagonals and Integer Area Constructions 

 
1M. Mahalakshmi, *2J. Kannan, 3A. Deepshika, 4Manju Somanath, 5P. Vijaya Shanthi and 

6K. Kaleeswari 
1,3,6Ph.D. Research Scholar, Department of Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to 

Madurai Kamaraj University), Sivakasi – 626124, Tamil Nadu, India. 

2Assistant Professor, Department of  Mathematics, Ayya Nadar Janaki Ammal College (Autonomous, affiliated to 

Madurai Kamaraj University), Sivakasi – 626124, Tamil Nadu, India. 

4Associate Professor, PG and Research Department of  Mathematics, National College (Autonomous, affiliated to 

Bharathidasan University), Trichy – 620 001, Tamil Nadu, India. 

5Assistant Professor, PG and Researh Department of  Mathematics, A.P.C. Mahalaxmi  College for Women (affiliated to 

Manonmaniam Sundaranar University), Thoothukudi – 628 002, Tamil Nadu, India. 

 

Article History: 

Received: 25-10-2024 

Revised: 02-12-2024 

Accepted: 12-12-2024 

Abstract:  

In this paper, we aim to collect all kites with certain fixed sides in integers, rational 

diagonals and integer areas. We develop mathematical procedure and python 

programming to collect all those kites. Also, experimental analyses are done by means 

of examples. 

Keywords: Diophantine equations, Geometrical shape, Integer area, Kite, Rational area, 

Rational diagonals. 

2010 MSC Subject Classification: 57N25, 97G30, 97F40, 11D09, 11D99. 

 

1. Introduction 

Geometry is one of the most interesting areas of mathematics. The basic geometry covers a range of 

shapes and deals with their properties along with the applications. In earlier days itself, 

Mathematicians involved themselves in finding some parameters regarding such shapes.   Some 

basic shapes are square, triangle, rectangle, and circle. Area and perimeter are the most used 

parameters by common people.  

To make the process of finding the area and perimeter much easier, early Mathematicians launched 

formulas using the terms which are needed to construct that particular shape. For example, if the side 

of a square is 𝑎, then its area is 𝑎2 and perimeter is 4𝑎.  So many regular shapes are seen along with 

those formulas.  

As an extension, researchers constructed some new shapes by adjoining existing known shapes. 

Also, they paved an unchallenging way to calculate its area. One such shape is Kite. It looks like the 

joining of two triangles under some considerations.  Using diagonals of kites, we can easily calculate 

its area. But here the task is to find the area of the kite by splitting it into two isosceles triangles, then 

find those triangle's area, and adding them up.  Heron's formula is used to find the area of those 

triangles, which is stated as ``If 𝑎, 𝑏 and 𝑐 are sides of a triangle, then its area is given as 

√𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where 𝑠 =
𝑎+𝑏+𝑐

2
 ".   



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In [14], (Rusin, 1998) collected rational triangles with equal area. Inspired by his work, this paper 

was developed. Also in [3], (Dickson, 2013) provided some key ideas for the creation of this 

problem. Also [ (Aassila, 2001), (Hendrik W. Lenstra, 2002), (Nelsen, 2020), (Robin McLean, 

1988), (Robin McLean, 1988), (Rusin, 1998), (Gopalan, Kannan, Manju, & Raja, 2016), (Manju, 

Kannan, & Raja, 2017), (Mahalakshmi, Kannan, & Narasimman, 2022), (Kannan & Mahalakshmi, 

Some Annotations on Almost And Pseudo Almost Equilateral Rational Rectangles, 2022)] are 

considered as roots for this paper. 

To do our work, we demand the need of classical branch of Mathematics, the Number Theory. 

Throughout this paper, the vast research area Diophantine Analysis, the study of Diophantine 

equations, its solutions and its applications, is employed (Dickson, 2013). Especially, the well-known 

Pythagorean equations and its solutions are used to collect the required kites.   

In this paper, we work on four types of kites (based on sides). In section 2, we display the 

preliminary concepts needed for this work. Section 3 collects all kites with sides 𝑛 and 𝑛 + 1 and 

having rational diagonal and integer area whereas in section 4, kites with sides 𝑛 and 𝑛 − 1 and with 

integer area are gathered.   In sections 5 and 6, we focus on the same but with sides 𝑛, 𝑛 ±  𝑟 

respectively. In each section, we include python coding to collect all required kites along with the 

output and experimental analysis by an example. 

2. Preliminaries 

Pythagorean equations are among the most pre-eminent Diophantine equations. These equations are 

of the form 𝑥2 + 𝑦2 = 𝑧2.  (Titu, Dorin, & Ion, 2010) Depicts that the general solution of the 

Pythagorean equation 𝑥2 + 𝑦2 = 𝑧2 over integers is of the form 𝑥 = 𝑘(𝑚2 − 𝑙2 ), 𝑦 = 2𝑘𝑚𝑙 and 

𝑧 = 𝑘(𝑚2 + 𝑙2) where 𝑘, 𝑚, 𝑙 ∈  ℤ and  𝑚 > 𝑙. 

A quadrilateral known as a kite has four sides that are grouped into two adjacent pairs of equal-

length sides, and the diagonals cross each other at right angles. 

 

Figure 1: Example of a Kite 

Note 2.1.: Throughout this paper, 𝑛 ∈  ℕ and 𝑟 ∈ ℕ − {1}. 

3. Kite with Sides 𝒏, 𝒏 + 𝟏  

 We address the kites with sides 𝑛 and 𝑛 + 1 in this part and gather all kites with rational diagonal 

and integer area by employing the Pythagorean equation and its solutions. 



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Let us consider a kite ABCD with sides 𝑛 and  𝑛 + 1 (Figure 2). 

 

Figure 2: Kite Used in Sections 3 and 5 

Take 𝐴𝐵 = 𝑛 = 𝐵𝐶 and 𝐴𝐷 = 𝑛 + 1 = 𝐷𝐶. Let  𝐴𝐶 = 𝑑. Then the kite 𝐴𝐵𝐶𝐷 can be split into two 

isosceles triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶. Thus 

Area of 𝐴𝐵𝐶𝐷 = Area of Δ 𝐴𝐵𝐶 +Area of Δ 𝐴𝐷𝐶 

The semi perimeter for Δ 𝐴𝐵𝐶 is 𝑠 =
𝑛+𝑛+𝑑

2
=

2𝑛+𝑑

2
.  Thus by applying Heron's formula, we obtain 

Area of  Δ 𝐴𝐵𝐶 =
𝑑

4
 √4𝑛2 − 𝑑2          (1) 

The semi perimeter for Δ 𝐴𝐷𝐶 is 𝑠 =
𝑛+1+𝑛+1+𝑑

2
=

2𝑛+2+𝑑

2
. Thus by applying Heron's formula, we 

obtain 

Area of  Δ 𝐴𝐷𝐶 =  
𝑑

4
 √(2𝑛 + 2)2 − 𝑑2         (2) 

This implies the fact that the Area of 𝐴𝐵𝐶𝐷 ∈ ℤ, if and only if the Area of Δ 𝐴𝐵𝐷 and Area of  

Δ 𝐴𝐷𝐶 both must be rational numbers. 

Area of 𝚫 𝑨𝑩𝑪 over ℚ 

``Equation (1)'' implies that Area of Δ 𝐴𝐵𝐶 ∈ ℚ if 4𝑛2 − 𝑑2 = 𝑡1
2  for some 𝑡1 ∈  ℚ . Now we 

reduced the problem of finding rational area for Δ 𝐴𝐵𝐶 to solve the equation 𝑡1
2 + 𝑑2 = (2𝑛)2 over 

ℚ. Let's first solve the equation 𝑡1
2 + 𝑑2 = 1 over ℚ to get started. 

Assume that 𝑡1 =
𝑝

𝑞
 and 𝑑 =

𝑟

𝑠
 for some 𝑝, 𝑞, 𝑟, 𝑠 ∈  ℤ . Then we acquire the Pythagorean type 

equation (𝑝𝑠)2 + (𝑞𝑟)2 = (𝑞𝑠)2. This equation has to be solved over integers.  

From section 2, we have 𝑡1 =
𝑝

𝑞
=

𝑚2−𝑙2

𝑚2+𝑙2
 and 𝑑 =

𝑟

𝑠
=

2𝑚𝑙

𝑚2+𝑙2
 for some 𝑚, 𝑙 ∈ ℤ . Hence the rational 

solutions for the equation 𝑡1
2 + 𝑑2 = (2𝑛)2 are 𝑡1 = 2𝑛 (

𝑚2−𝑙2

𝑚2+𝑙2
) and 𝑑 = 4𝑛 (

𝑚𝑙

𝑚2+𝑙2
) for some 𝑚, 𝑙 ∈

 ℤ  and 𝑚 > 𝑙. 

 

 



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Area of 𝚫 𝑨𝑫𝑪 over ℚ 

``Equation (2)'' implies that Area of Δ 𝐴𝐷𝐶 ∈ ℚ if (2𝑛 + 2)2 − 𝑑2 = 𝑡2
2  for some 𝑡2 ∈  ℚ . As in the 

above case, we have 𝑡2 = (2𝑛 + 2) (
𝑎2−𝑏2

𝑎2+𝑏2
)   and 𝑑 = (2𝑛 + 2) (

2𝑎𝑏

𝑎2+𝑏2
 )  for some 𝑎, 𝑏 ∈ ℤ and 𝑎 >

𝑏. 

To find required 𝒏 

Since the value 𝑑 is equal in both the areas, we can equate them and find 𝑛 as 

𝑛 =
𝑎𝑏(𝑚2 + 𝑙2)

𝑚𝑙(𝑎2 + 𝑏2) − 𝑎𝑏(𝑚2 + 𝑙2)
 

3.1. Python coding for generation of required kites with sides 𝒏 and 𝒏 + 𝟏 

 

In this section we display python coding to collect kites with sides 𝑛, 𝑛 + 1, rational diagonals and 

integer area followed by its output. 

Figure 3: Coding 1: Kite with sides 𝑛, 𝑛 +  1 

 



Communications on Applied Nonlinear Analysis 

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Figure 4: Output: Coding 1 

3.2. Experimental Analysis by an Example 

Usual formula for area of kite is 
1

2
𝑑𝑑1, where 𝑑 and 𝑑1  are diagonals. In figure 1, take the diagonals 

as 𝐵𝐷(= 𝑑), 𝐴𝐶(= 𝑑1) and the sides as 𝐴𝐷 = 𝑛 = 𝐴𝐵 and 𝐵𝐶 = 𝑛 + 1 = 𝐶𝐷. Applying Pythagoras 

theorem for Δ 𝐴𝑂𝐵 and  Δ 𝐵𝑂𝐶, we get 𝑂𝐴 =
1

2
√4𝑛2 − 𝑑2 and 𝑂𝐶 =

1

2
√4(𝑛 + 1)2 − 𝑑2 respectively.  

This gives 𝑑1 = 𝑂𝐴 + 𝑂𝐶 =
1

2
[√4𝑛2 − 𝑑2 + √4(𝑛 + 1)2 − 𝑑2 ].  

So the area is 
1

4
𝑑 [√4𝑛2 − 𝑑2 + √4(𝑛 + 1)2 − 𝑑2 ]. 

Now, let us verify this by an example. Take 𝑛 = 20 and 𝑑 =
840

29
. Then we obtain 𝑑1 = 29. This gives 

the area 420, is same as the one found from the python coding. 

4. Kite with sides 𝒏, 𝒏 − 𝟏 

Utilizing the Pythagorean equation and its solutions, we examine the kites with sides 𝑛 and 𝑛 − 1 in 

this section and generate all kites with integer area.  

Let us consider a kite 𝐴𝐵𝐶𝐷 with sides 𝑛 and 𝑛 − 1 (Figure 5).  

Take 𝐴𝐵 = 𝑛 = 𝐵𝐶 and 𝐴𝐷 = 𝑛 − 1 = 𝐷𝐶. Let 𝐴𝐶 = 𝑑. Then the kite 𝐴𝐵𝐶𝐷 can be split into two 

isosceles triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶.  

Thus,          Area of 𝐴𝐵𝐶𝐷 = Area of  Δ 𝐴𝐵𝐶 + Area of Δ 𝐴𝐷𝐶 



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Figure 5: Kite Used in Sections 4 and 6 

The semi perimeter for Δ 𝐴𝐵𝐶 is 𝑠 =
𝑛+𝑛+𝑑

2
=

2𝑛+𝑑

𝑑
 . Thus by applying Heron's formula, we obtain 

Area of  Δ 𝐴𝐵𝐶 =
𝑑

4
 √4𝑛2 − 𝑑2          (3) 

The semi perimeter for Δ 𝐴𝐷𝐶 is 𝑠 =
𝑛−1+𝑛−1+𝑑

2
=

2𝑛−2+𝑑

2
. Thus by applying Heron's formula, we 

obtain 

Area of  Δ 𝐴𝐷𝐶 =  
𝑑

4
 √(2𝑛 − 2)2 − 𝑑2         (4) 

This implies the fact that the Area of 𝐴𝐵𝐶𝐷 ∈ ℤ, if and only if the Area of Δ 𝐴𝐵𝐷 and Area of  

Δ 𝐴𝐷𝐶 both must be rational numbers. 

Area of 𝚫 𝑨𝑩𝑪 over ℚ 

``Equation (3)'' implies that Area of Δ 𝐴𝐵𝐶 ∈ ℚ if 𝑡1 = 2𝑛 (
𝑚2−𝑙2

𝑚2+𝑙2
)  and 𝑑 = 4𝑛 (

𝑚𝑙

𝑚2+𝑙2
) for some 

𝑚, 𝑙 ∈  ℤ  and 𝑚 > 𝑙. 

 Area of 𝚫 𝑨𝑫𝑪 over ℚ 

``Equation (4)'' implies that Area of Δ 𝐴𝐷𝐶 ∈ ℚ if (2𝑛 − 2)2 − 𝑑2 = 𝑡2
2  for some 𝑡2 ∈  ℚ . As in the 

above case, we have 𝑡2 = (2𝑛 − 2)
(𝑎2−𝑏2)

(𝑎2+𝑏2)
  and 𝑑 = (2𝑛 − 2) (

2𝑎𝑏

𝑎2+𝑏2
 )  for some 𝑎, 𝑏 ∈ ℤ and 𝑎 > 𝑏. 

To find required n 

Since the value 𝑑 is equal in both the areas, we can equate them and find 𝑛 as 

𝑛 =
𝑎𝑏(𝑚2 + 𝑙2)

𝑎𝑏(𝑚2 + 𝑙2) − 𝑚𝑙(𝑎2 + 𝑏2)
 

4.1. Python coding for generation of required kites with sides 𝒏 and 𝒏 − 𝟏 

In this section we display python coding to collect kites with sides 𝑛, 𝑛 − 1, rational diagonals and 

integer area followed by its output. 



Communications on Applied Nonlinear Analysis 

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Figure 6: Coding 2: Kite with sides 𝑛, 𝑛 − 1

 

Figure 7: Output: Coding 2 

 

4.2. Experimental Analysis by an Example 

In figure 1, take the diagonals as 𝐵𝐷(= 𝑑), 𝐴𝐶(= 𝑑1) and the sides as 𝐶𝐵 = 𝑛 = 𝐶𝐷 and 𝐴𝐷 = 𝑛 −

1 = 𝐴𝐵. Doing the same procedure as in subsection 3.2, we get    

 𝑑1 =
1

2
[√4𝑛2 − 𝑑2 + √4(𝑛 − 1)2 − 𝑑2 ]. So the area is 

1

4
𝑑 [√4𝑛2 − 𝑑2 + √4(𝑛 − 1)2 − 𝑑2 ]. 

Now, let us verify this by an example. Take 𝑛 = 4 and 𝑑 =
24

5
. Then we obtain 𝑑1 = 5. This gives the 

area 12, is same as the one found from the python coding. 



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5. Kite with Sides 𝒏, 𝒏 + 𝒓  

In this section, we work on kites with sides 𝑛 and 𝑛 + 𝑟, as above. Let us consider a kite A𝐵𝐶𝐷 with 

sides 𝑛 and 𝑛 + 𝑟. In fig 2, take 𝐴𝐵 = 𝑛 = 𝐵𝐶 and 𝐴𝐷 = 𝑛 + 𝑟 = 𝐷𝐶. Let 𝐴𝐶 = 𝑑. Then the kite 

𝐴𝐵𝐶𝐷 can be split into two isosceles triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶.  

Thus, 

Area of 𝐴𝐵𝐶𝐷 = Area of Δ 𝐴𝐵𝐶 +Area of Δ 𝐴𝐷𝐶 

As in the previous sections, we get  Δ 𝐴𝐵𝐶  

Area of  Δ 𝐴𝐵𝐶 =
𝑑

4
 √4𝑛2 − 𝑑2          (5) 

The semi perimeter for Δ 𝐴𝐷𝐶 is 𝑠 =
𝑛+𝑟+𝑛+𝑟+𝑑

2
=

2𝑛+2𝑟+𝑑

2
. Thus by applying Heron's formula, we 

obtain, 

Area of  Δ 𝐴𝐷𝐶 =  
𝑑

4
 √(2𝑛 + 2𝑟)2 − 𝑑2         (6) 

This implies the fact that the Area of 𝐴𝐵𝐶𝐷 ∈ ℤ, if and only if the Area of Δ 𝐴𝐵𝐷 and Area of  

Δ 𝐴𝐷𝐶 both must be rational numbers. 

Area of 𝚫 𝑨𝑩𝑪 over ℚ 

``Equation (5)'' implies that Area of Δ 𝐴𝐵𝐶 ∈ ℚ if  𝑡1 = 2𝑛 (
𝑚2−𝑙2

𝑚2+𝑙2
) and 𝑑 = 4𝑛 (

𝑚𝑙

𝑚2+𝑙2
) for some 

𝑚, 𝑙 ∈  ℤ  and 𝑚 > 𝑙. 

Area of 𝚫 𝑨𝑫𝑪 over ℚ 

``Equation (6)'' implies that Area of Δ 𝐴𝐷𝐶 ∈ ℚ if (2𝑛 + 2𝑟)2 − 𝑑2 = 𝑡2
2  for some 𝑡2 ∈  ℚ . As in 

the above case, we have 𝑡2 = (2𝑛 + 2𝑟) (
𝑎2−𝑏2

𝑎2+𝑏2
)    and 𝑑 = (2𝑛 + 2𝑟) (

2𝑎𝑏

𝑎2+𝑏2
 )  for some 𝑎, 𝑏 ∈ ℤ 

and 𝑎 > 𝑏. 

To find required 𝒏 

Since the value 𝑑 is equal in both the areas, we can equate them and find 𝑛 as 

𝑛 =
𝑟𝑎𝑏(𝑚2 + 𝑙2)

𝑚𝑙(𝑎2 + 𝑏2) − 𝑎𝑏(𝑚2 + 𝑙2)
 

5.1. Python coding for generation of required kites with sides 𝒏 and 𝒏 + 𝒓 

In this section we display python coding to collect kites with sides 𝑛, 𝑛 + 𝑟, rational diagonals and 

integer area followed by its output. 



Communications on Applied Nonlinear Analysis 

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Figure 8: Coding 3: Kite with sides 𝑛, 𝑛 +  𝑟 

 

Figure 9: Output 3: Kite with sides 𝑛, 𝑛 +  𝑟 

5.2. Experimental Analysis by an Example 

In figure 1, take the diagonals as 𝐵𝐷(= 𝑑), 𝐴𝐶(= 𝑑1) and the sides as 𝐴𝐷 = 𝑛 = 𝐴𝐵 and 𝐵𝐶 = 𝑛 +

𝑟 = 𝐶𝐷. Doing the same procedure as in subsection 3.2., we get 

This gives 𝑑1 =
1

2
[√4𝑛2 − 𝑑2 + √4(𝑛 + 𝑟)2 − 𝑑2 ].  

So the area is 
1

4
𝑑 [√4𝑛2 − 𝑑2 + √4(𝑛 + 𝑟)2 − 𝑑2 ]. 



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Now, let us verify this by an example. Take 𝑛 = 50, 𝑟 = 10 and 𝑑 = 96. Then we obtain 𝑑1 = 50. 

This gives the area 2400, is same as the one found from the python coding. 

6. Kite with sides 𝒏, 𝒏 − 𝒓 

In this section, we on kites with sides  𝑛 − 𝑟, as above. Let us consider a kite 𝐴𝐵𝐶𝐷 with sides 𝑛 and 

𝑛 − 𝑟. In figure 5, take 𝐴𝐵 = 𝑛 = 𝐵𝐶 and 𝐴𝐷 = 𝑛 − 𝑟 = 𝐷𝐶.  

Let 𝐴𝐶 = 𝑑. Then the kite 𝐴𝐵𝐶𝐷 can be split into two isosceles triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐶.  

Thus,          Area of 𝐴𝐵𝐶𝐷 = Area of  Δ 𝐴𝐵𝐶 + Area of Δ 𝐴𝐷𝐶 

As in previous sections, we get Area of  Δ 𝐴𝐵𝐶 =
𝑑

4
 √4𝑛2 − 𝑑2      (7) 

   and 

Area of  Δ 𝐴𝐷𝐶 =  
𝑑

4
 √(2𝑛 + 2𝑟)2 − 𝑑2         (8) 

This implies the fact that the Area of 𝐴𝐵𝐶𝐷 ∈  ℤ, if and only if the Area of  Δ 𝐴𝐵𝐷 and Area of 

Δ 𝐴𝐷𝐶 both must be rational numbers. 

Area of 𝚫 𝑨𝑩𝑪 over ℚ 

``Equation (7)'' implies that Area of Δ 𝐴𝐵𝐶 ∈ ℚ if  𝑡1 = 2𝑛 (
𝑚2−𝑙2

𝑚2+𝑙2
) and 𝑑 = 4𝑛 (

𝑚𝑙

𝑚2+𝑙2
) for some 

𝑚, 𝑙 ∈  ℤ  and 𝑚 > 𝑙. 

Area of 𝚫 𝑨𝑫𝑪 over ℚ 

``Equation (8)'' implies that Area of Δ 𝐴𝐷𝐶 ∈ ℚ if (2𝑛 − 2𝑟)2 − 𝑑2 = 𝑡2
2  for some 𝑡2 ∈  ℚ .  

As in the above case, we have 𝑡2 = (2𝑛 − 2𝑟)
(𝑎2−𝑏2)

(𝑎2+𝑏2)
  and 𝑑 = (2𝑛 − 2𝑟) (

2𝑎𝑏

𝑎2+𝑏2
 )  for some 𝑎, 𝑏 ∈ ℤ 

and 𝑎 > 𝑏. 

To find required n 

Since the value 𝑑 is equal in both the areas, we can equate them and find 𝑛 as 

𝑛 =
𝑟𝑎𝑏(𝑚2 + 𝑙2)

𝑎𝑏(𝑚2 + 𝑙2) − 𝑚𝑙(𝑎2 + 𝑏2)
 

6.1. Python coding for generation of required kites with sides 𝒏 and 𝒏 − 𝒓 

In this section we display python coding to collect kites with sides 𝑛, 𝑛 −  𝑟 

 rational diagonals and integer area followed by its output. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

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https://internationalpubls.com 

 

Figure 10: Coding 4: Kite with sides 𝑛, 𝑛 −  𝑟 

 

Figure 11: Output 4: Kite with sides 𝑛, 𝑛 −  𝑟 

6.2. Experimental Analysis by an Example 

In figure 1, take the diagonals as 𝐵𝐷(= 𝑑), 𝐴𝐶(= 𝑑1) and the sides as 𝐶𝐵 = 𝑛 = 𝐶𝐷 and AD= 𝑛 −

𝑟 = 𝐴𝐵. Doing the same procedure as in subsection 3.2., we get 

This gives 𝑑1 =
1

2
[√4𝑛2 − 𝑑2 + √4(𝑛 − 𝑟)2 − 𝑑2 ].  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

12 
https://internationalpubls.com 

So the area is 
1

4
𝑑 [√4𝑛2 − 𝑑2 + √4(𝑛 − 𝑟)2 − 𝑑2 ]. 

Now, let us verify this by an example. Take 𝑛 = 375, 𝑟 = 6 and 𝑑 = 720. Then we obtain 𝑑1 = 186. 

This gives the area 66960, is same as the one found from the python coding. 

7. Conclusion 

In this paper, some particular choices of kites with sides 𝑛, 𝑛 ± 1 and 𝑛, 𝑛 ±  𝑟 having integer area 

and rational diagonal are collected mathematically as well as through Python programming. These 

types of concepts may be applied in the field of architecture. In future, this thought may extend to 

other geometrical shapes also.   

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