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Math & Conversions Ellipse Calculator 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: $A x^2 + C y^2 + D x + E y + F = 0$ A C D E F B ($x y$ term, assumed 0) Calculate Ellipse Properties 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: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ 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: $$c^2 = (\text{larger denominator}) - (\text{smaller denominator})$$ $$\text{or } c^2 = a^2 - b^2$$ 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) What is the relationship between a, b, and c in an ellipse? In an ellipse, '$a$' is always the semi-major axis (the larger of the two), '$b$' is the semi-minor axis (the smaller), and '$c$' is the distance from the center to a focus. They are related by the formula $c^2 = a^2 - b^2$ (or $a^2 = b^2 + c^2$). Note that unlike a hyperbola, '$a$' is *not* always the x-semiaxis. What is the general equation of an ellipse? The most general form is a conic section where $A \neq C$ but share the same sign: $A x^2 + C y^2 + D x + E y + F = 0$. The simpler, standard form (canonical) is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. Is there an exact formula for the circumference of an ellipse? No, the circumference of an ellipse cannot be calculated with a simple, exact algebraic formula. It requires an elliptical integral of the second kind. For practical purposes, approximations are used, such as Ramanujan's approximation, which is highly accurate: $C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$. What is the eccentricity of an ellipse? Eccentricity ($e$) measures how much an ellipse deviates from a perfect circle ($e=0$). It is calculated as $e = c/a$. For an ellipse, $0 \le e < 1$. The higher the eccentricity, the more elongated the ellipse. Key Ellipse Formulas Foci Relationship $$c^2 = a^2 - b^2$$ Area $$Area = \pi ab$$ Eccentricity $$e = \frac{c}{a}$$ Related Conic Section Tools Conic Classifier Hyperbola Calculator Parabola Calculator Circle Calculator Area Calculator (General)
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