Calculating the area of an ellipse in C++

Ellipses are geometric shapes defined by their unique properties and play a vital role in mathematical and real-world applications. This article helps to calculate the area of the ellipse in C++.

An ellipse is a closed curve with different features apart from other geometric shapes. Unlike circles, ellipses have two distinct axes: a major axis and a minor axis. The major axis is the longest diameter, while the minor axis is the shortest in the ellipse. In a plane, an ellipse is symmetric with respect to two perpendicular axes. It is one of the five conic sections formed by the intersection of a plane with a cone. An ellipse can be defined as the locus of points in a plane, the sum of whose distances from two fixed points is a const. The two fixed points are called the foci of the ellipse. The distance between the foci is called the focal length.

Mathematical Formula for ellipse Area:

The formula will change when the given parameters change.

If the length of the semi-major axis and the length of the semi-minor axis are given, the formula for finding the area will be πab where a is the semi-major axis length, and b is the semi-minor axis length.

If the length of the semi-major axis length is given and eccentric is given, the area will be a^2π√(1-e^2).

Where a is the semi-major axis length, and e is the eccentric.

When the implicit equation for the ellipse is given, which is in the form of Ax^2 + Bxy+ Cy^2 =1, then the area will be 2π/(√(4AC - B^2)).

If the ellipse is centered at the origin, the positive axis intercepts are x and y. The extreme right and top coordinates are xm, and ym will be

π xm y=π x ym.

Example:

Let us take a C++ program for calculating area when the semi-major axis and semi-minor axis are given.

#include<iostream>
#include<cmath>
using namespace std;
int main() {
 double majorAxis, minorAxis;
 cout << "Enter the semi major axis length: ";
 cin >> majorAxis;
 cout << "Enter the semi-minor axis length: ";
 cin >> minorAxis;
 double area = M_PI * majorAxis * minorAxis;
 cout << "The area of the ellipse is: " << area << endl;
 return 0;
}

Output:

Calculating the area of an ellipse in C++

Explanation:

This C++ program calculates and displays the area of an ellipse using the lengths of its semi-major and semi-minor axes. The user is prompted to input these values, and the program uses the formula πab to compute the area.

Example 2:

Let us take a C++ program to calculate the area of the ellipse when eccentricity and semi major axis are given.

#include<iostream>
#include<cmath>
using namespace std;
int main() {
 double a, e;
 cout << "Enter the length of semi-major axis (a): ";
 cin >> a;
 cout << "Enter the eccentricity (e): ";
 cin >> e;
 double area = a * a * M_PI * sqrt(1 - e * e);
 cout << "The area of the ellipse is: " << area << endl;
 return 0;
}

Output:

Explanation:

This concise C++ program calculates and displays the area of an ellipse using the length of its semi-major axis (a) and eccentricity (e). User input is used to gather these values, and a formula is applied to compute the area.

Example 3:

Let us take a C++ program when the equation is given.

#include<iostream>
#include<cmath>
using namespace std;
int main() {
 // Equation of the ellipse is Ax^2 + Bxy+ Cy^2 = 1
 double A, B, C;
 // Prompt the user to input the coefficients of the implicit equation
 cout << "Enter the coefficient A: ";
 cin >> A;
 cout << "Enter the coefficient B: ";
 cin >> B;
 cout << "Enter the coefficient C: ";
 cin >> C;
 double area = 2 * M_PI / sqrt(4 * A * C - B * B);
 cout << "The area of the ellipse is: " << area << endl;
 return 0;
}

Output:

Explanation:

This program will take the coefficients in the equation from the user and use them in the formula to calculate the area of an ellipse.

Example 4:

Let us take a C++ program when intercepts are given.

#include<iostream>
#include<cmath>
using namespace std;
int main() {
 double x, y, xm, ym;
 cout << "Enter positive x-axis intercept (x): ";
 cin >> x;
 cout << "Enter positive y-axis intercept (y): ";
 cin >> y;
 cout << "Enter extreme right coordinate (xm): ";
 cin >> xm;
 cout << "Enter extreme top coordinate (ym): ";
 cin >> ym;
 double area = M_PI * xm * y;
 cout << "The area of the ellipse is: " << area << endl;
 return 0;
}

Output:

Explanation:

Feel free to copy and use this C++ program to calculate the area of an ellipse based on its positive x-axis intercept, positive y-axis intercept, extreme right coordinate, and extreme top coordinate.

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