Area of Circumcircle and Incircle of a Right Kite

Given two positive integers A and B representing the sides of the right kite, the task is to find the area of the circumcircle and incircle of a right kite.

A right kite is a kite that can be inscribed in a circle with two opposite angles are at right angles. The line of symmetry of the kite is also the diameter of the circumcircle of the kite. It divides the kite into two congruent right-angled triangles having sides as A and B of a right kite.

Examples:

Input: A = 3, B = 4
Output: Area of circumcircle of Right Kite is 19.625, Area of incircle of Right Kite is 3.14

Input: A = 10, B = 5
Output: Area of circumcircle of Right Kite is 98.125, Area of incircle of Right Kite is 28.26

Approach: There are some observations to solve this problem. Follow the steps below to solve this problem:

• Here, a = AB = AD and b = BC = CD
• In the kite ABCD with opposite angles B and D as 90°, thus the opposite angles can be calculated as tan (A/2) = b/a and tan(C/2) = a/b
• Let p as the length of diagonal AC and q as the length of diagonal BD.
• Diagonal AC can be easily calculated using the Pythagoras Theorem. Hence p = (a² + b²)^½
• Since the diagonal is equal to the diameter of the circumcircle of the kite, the radius of the circumcircle is calculated as R = (a² + b²)^½/2
• Thus, the area of the circumcircle will be pi * R* R 
• Also, all kites are tangential quadrilaterals, therefore the radius of the incircle can be calculated by r = Area of kite/Semiperimeter of the kite i.e r = a*b/(a+b).
• Thus, the area of the incircle will be pi*r*r.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;
#define pi 3.14

// Function to calculate the area of
// circumcircle of right kite
double AreaOfCircumcircle(int a, int b)
{
    // Find the radius
    double radius = sqrt(a * a + b * b)
                    / 2;
    return pi * radius * radius;
}

// Function to calculate the area of
// incircle of right kite
double AreaOfIncircle(int a, int b)
{
    // Find the radius
    double radius = (a * b) / (a + b);

    return pi * radius * radius;
}

// Driver Code
int main()
{
    // Given Input
    int a, b;
    a = 10;
    b = 5;

    // Function Call
    double circumarea = AreaOfCircumcircle(
        a, b);
    cout << "Area of circumcircle of Right Kite is"
         << " " << circumarea << endl;

    // Function Call
    double inarea = AreaOfIncircle(
        a, b);
    cout << "Area of incircle of Right Kite is"
         << " " << inarea << endl;

    return 0;
}

Java

// Java program for the above approach
public class GFG {
    static double pi = 3.14;

    // Function to calculate the area of
    // circumcircle of right kite
    static double AreaOfCircumcircle(int a, int b)
    {
        // Find the radius
        double radius = Math.sqrt(a * a + b * b) / 2;
        return pi * radius * radius;
    }

    // Function to calculate the area of
    // incircle of right kite
    static double AreaOfIncircle(int a, int b)
    {
        // Find the radius
        double radius = (a * b) / (a + b);

        return pi * radius * radius;
    }

    // Driver code
    public static void main(String[] args)
    {

        // Given Input
        int a, b;
        a = 10;
        b = 5;

        // Function Call
        double circumarea = AreaOfCircumcircle(a, b);
        System.out.printf(
            "Area of circumcircle of Right Kite is %.3f\n",
            circumarea);

        // Function Call
        double inarea = AreaOfIncircle(a, b);
        System.out.printf(
            "Area of incircle of Right Kite is %.2f\n",
            inarea);
    }
}

// This code is contributed by abhinavjain194

Python3

# Python program for the above approach
# Function to calculate the area of
# circumcircle of right kite
import math
pi = 3.14

def AreaOfCircumcircle(a, b):

    # Find the radius
    radius = math.sqrt(a * a + b * b)/ 2
    return pi * radius * radius


# Function to calculate the area of
# incircle of right kite
def AreaOfIncircle( a,  b):

    # Find the radius
    radius = (a * b) // (a + b)
    return pi * (radius**2)


# Driver Code
# Given Input
a = 10
b = 5

# Function Call
circumarea = AreaOfCircumcircle(a, b)
print("Area of circumcircle of Right Kite is" ," " , format(circumarea,".3f"))

# Function Call
inarea = AreaOfIncircle(a, b)
print("Area of incircle of Right Kite is" ," " , format(inarea,".2f"))

# this code is contributed by shivanisinghss2110

C#

// C# program for the above approach
using System;

class GFG{

static double pi = 3.14;

// Function to calculate the area of
// circumcircle of right kite
static double AreaOfCircumcircle(int a, int b)
{

    // Find the radius
    double radius = Math.Sqrt(a * a + b * b) / 2;
    return pi * radius * radius;
}

// Function to calculate the area of
// incircle of right kite
static double AreaOfIncircle(int a, int b)
{

    // Find the radius
    double radius = (a * b) / (a + b);

    return pi * radius * radius;
}

// Driver code
public static void Main()
{

    // Given Input
    int a, b;
    a = 10;
    b = 5;

    // Function Call
    double circumarea = AreaOfCircumcircle(a, b);
    Console.WriteLine(
        "Area of circumcircle of Right Kite is " +
        circumarea);

    // Function Call
    double inarea = AreaOfIncircle(a, b);
    Console.WriteLine(
        "Area of incircle of Right Kite is " + inarea);
}
}

// This code is contributed by subhammahato348

Javascript

<script>

        // JavaScript Program for the above approach

        var pi = 3.14

        // Function to calculate the area of
        // circumcircle of right kite
        function AreaOfCircumcircle(a, b) {
            // Find the radius
            let radius = Math.sqrt(a * a + b * b)
                / 2;
            return pi * radius * radius;
        }

        // Function to calculate the area of
        // incircle of right kite
        function AreaOfIncircle(a, b) {
            // Find the radius
            let radius = (a * b) / (a + b);

            return pi * radius * radius;
        }

        // Driver Code

        // Given Input
        let a, b;
        a = 10;
        b = 5;

        // Function Call
        let circumarea = AreaOfCircumcircle(a, b);
        document.write("Area of circumcircle of Right Kite is " +
         (circumarea.toFixed(3)) + "<br>");

        // Function Call
        let inarea = AreaOfIncircle(
            a, b);
        document.write("Area of incircle of Right Kite is " +
         (inarea.toFixed(3)) + "<br>");


    // This code is contributed by Potta Lokesh

</script>

Output

Area of circumcircle of Right Kite is 98.125
Area of incircle of Right Kite is 28.26

Time Complexity: O(logn) since using inbuilt sqrt function
Auxiliary Space: O(1)