$$A=\pi\space ab$$


The formula for the area of an ellipse, \(A = a \times b \times \pi\), where \(a\) and \(b\) are the semi-major and semi-minor axes respectively, multiplied by \(\pi\), is derived from geometric principles and the characteristics of an ellipse.

Here's an explanation for why \(a \times b \times \pi\) represents the area of an ellipse:

  1. Understanding the Ellipse:
    An ellipse is a closed curve in a plane, resembling a squashed circle, where the sum of the distances from any point on the curve to two fixed points (the foci) is constant. The major axis is the longest diameter of the ellipse, while the minor axis is the shortest.
  2. Derivation of the Formula:
    The area of an ellipse is somewhat more complex to derive compared to a circle, but it can be understood by utilizing its resemblance to a circle. Consider the formula for the area of a circle: \(A = \pi r^2\), where \(r\) is the radius.
  3. Similarities with a Circle:
    In the case of a circle, the radius is the same in all directions, making the area calculation relatively simple: \(A = \pi r^2\).
  4. Elliptical Nature:
    For an ellipse, however, the situation is different because it has different lengths for the semi-major axis (\(a\)) and the semi-minor axis (\(b\)).
  5. Application of Proportions:
    The area of an ellipse is based on the lengths of its axes, but unlike a circle, the values of \(a\) and \(b\) are different. Hence, to calculate the area, the formula involves both axes and the constant factor \(\pi\).
  6. The Formula for the Area:
    The formula \(A = a \times b \times \pi\) is used to calculate the area of an ellipse, where:
    • \(a\) is the length of the semi-major axis.
    • \(b\) is the length of the semi-minor axis.
    • \(\pi\) is a mathematical constant approximately equal to 3.14159.

The area of an ellipse is fundamentally different from that of a circle due to the elongation and compression along the axes, and hence the formula \(A = a \times b \times \pi\) considers both axes' lengths and incorporates the factor \(\pi\) to determine the overall area enclosed by the ellipse.