The formula for the area of an ellipse, \(A = a \times b \times \pi\),
where \(a\) and \(b\) are the semimajor and semiminor 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:

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.

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.

Similarities with a Circle:
In the case of a circle, the radius is the same in all
directions, making the area calculation relatively simple: \(A =
\pi r^2\).

Elliptical Nature:
For an ellipse, however, the situation is different because it
has different lengths for the semimajor axis (\(a\)) and the
semiminor axis (\(b\)).

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 \(\pi\).

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 semimajor axis.
 \(b\) is the length of the semiminor 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.