Ellipse Calculator (Conic Sections)
Calculate all the geometric and analytical properties of an ellipse. Enter the coefficients from the general form of the conic section: $\mathbf{A x^2 + C y^2 + D x + E y + F = 0}$ where $A$ and $C$ must be positive and not equal.
General Form: $A x^2 + C y^2 + D x + E y + F = 0$
Key Properties
Center $(h, k)$
Standard Form Equation
Axes, Foci, and Area
Semi-Major Axis ($a$)
Semi-Minor Axis ($b$)
Foci Distance ($c$)
Eccentricity ($e$)
Area
Approximate Circumference (Ramanujan)
Step-by-Step (Completing the Square)
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: $$C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$$