Ellipse Calculator (Conic Sections)

Solve ellipse properties from center, axes, foci, area, and circumference inputs.

The Standard Form of an Ellipse

The standard (canonical) form of an ellipse centered at $(h, k)$ is essential for identifying its properties quickly. The equation is:

Where:

• $(h, k)$: The coordinates of the center.
• $a$: The semi-major or semi-minor axis associated with the $x$ terms.
• $b$: The semi-major or semi-minor axis associated with the $y$ terms.

The semi-major axis (the larger of $a$ and $b$) determines the orientation of the ellipse (horizontal if under $x$, vertical if under $y$).

Finding the Foci and Eccentricity

The foci (plural of focus) are two fixed points inside the ellipse used to define its shape. The distance from the center to each focus is denoted by $c$, and it is related to the semi-axes by the Pythagorean-like relationship:

The **eccentricity** ($e$) measures the flatness of the ellipse and is defined as $e = \frac{c}{a}$ (where $a$ is the length of the semi-major axis). For an ellipse, $0 \le e < 1$.

Area and Circumference

Unlike a circle, the area calculation for an ellipse is simple, but its circumference is complex.

• Area: The area enclosed by the ellipse is exactly equal to $\pi$ times the product of the two semi-axes: $$\text{Area} = \pi ab$$
• Circumference (Perimeter): The circumference $C$ has no exact algebraic solution. We use the highly accurate approximation formula discovered by Srinivasa Ramanujan: $$C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$$

Frequently Asked Questions (FAQ)

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\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

c^2 = (\text{larger denominator}) - (\text{smaller denominator})

\text{or } c^2 = a^2 - b^2

\text{Area} = \pi ab

C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

$c^2 = (\text{larger denominator}) - (\text{smaller denominator})$ $\text{or } c^2 = a^2 - b^2$