Perimeter and Area of an Ellipse

You might expect that there is a simple, closed-form solution to find the perimeter of an ellipse. However, without the application of calculus, there is no way to exactly solve for the perimeter of an ellipse. Many attempts have been made, however, and two of the approximate solutions are given below.

The variables you will need to solve this problem are represented graphically in Figure 1, below:

Figure 1. Ellipse Equation Variables.

The first solution is very simple, but it is not as accurate as the second. To determine the approximate perimeter of the ellipse, simply apply the following equation:

where a is one-half the major axis, and b is one-half the minor axis.

The second solution is slightly more complex, but it is especially accurate. The same variables from above are used in the following equation:

Fortunately, there is a closed-form solution for the area of an ellipse that provides an exact solution. To determine the area of an ellipse, simply apply the following equation: