Area of an Ellipse

The ellipse x²/a² + y²/b² = 1 encloses area πab. Reduces to πa² for a circle (b = a). Derived by scaling the unit circle uniformly in one direction.

Method 1 — Scaling argument:

The ellipse $x^2/a^2 + y^2/b^2 = 1$ is obtained from the circle $x^2 + y^2 = a^2$ by the transformation $(x, y) \mapsto (x, by/a)$, which scales all $y$-coordinates by $b/a$.

A uniform scaling in one direction scales all areas by the same factor. Therefore:

$$\text{Area of ellipse} = \frac{b}{a} \times \text{Area of auxiliary circle} = \frac{b}{a} \times \pi a^2 = \pi ab$$

Method 2 — Integration:

For the upper half of the ellipse, $y = b\sqrt{1 - x^2/a^2}$. Total area:

$$A = 2\int_{-a}^{a} b\sqrt{1-\frac{x^2}{a^2}}\,dx$$

Substitute $x = a\sin\theta$, $dx = a\cos\theta\,d\theta$:

$$A = 2\int_{-\pi/2}^{\pi/2} b\cos\theta \cdot a\cos\theta\,d\theta = 2ab\int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta$$ $$= 2ab \cdot \frac{\pi}{2} = \pi ab$$

Special cases: $a = b = r$ gives $\pi r^2$ (circle); $b \to 0$ gives area $\to 0$ (degenerate ellipse collapses to a line segment).