Area of Ellipse

We can simplify the geometry by using a linear affine transformation to scale the ellipse into a unit circle. Let: $$x' = \frac{x}{a}, \quad y' = \frac{y}{b}$$
 The ellipse equation transforms directly into a standard unit circle equation:
$$(x')^2 + (y')^2 = 1$$
 Therefore, the area relationship between the original plane and the transformed circular plane is:
$$\text{Area}(\text{Ellipse Triangle}) = ab \cdot \text{Area}(\text{Circle Triangle})$$
In a unit circle ($$R = 1$$), the inscribed polygon that maximizes area is a regular equilateral triangle. The side length of an equilateral triangle inscribed in a unit circle is $$s = \sqrt{3}$$.
$$\text{Area}' = \frac{\sqrt{3}}{4}s^2 = \frac{\sqrt{3}}{4}(\sqrt{3})^2 = \frac{3\sqrt{3}}{4}$$
$$\text{Maximum Area} = \frac{3\sqrt{3}}{4}ab$$