Find the center of an ellipse from the general equation Ax²+Bxy+Cy²+Dx+Ey+F=0 or standard form parameters. Computes center, semi-axes, eccentricity, foci, area, and circumference.
An ellipse is the set of all points in a plane where the sum of the distances to two fixed points (the foci) is constant. Every ellipse has a center — the midpoint between its foci — around which it is symmetric. The standard form of an ellipse centered at (h, k) is (x−h)²/a² + (y−k)²/b² = 1, where a and b are the semi-major and semi-minor axes.
Often, however, an ellipse is given in general form: Ax² + Bxy + Cy² + Dx + Ey + F = 0. To find the center and axis lengths from this equation, you complete the square for x and y, grouping and factoring until the equation matches standard form. The center is then (h, k) = (−D/2A, −E/2C) when B = 0 (no rotation).
Knowing the center unlocks many other properties. The eccentricity e = c/a, where c = √(a²−b²), measures how "stretched" the ellipse is — 0 for a circle, approaching 1 for a very elongated shape. The foci lie along the major axis at distance c from the center. The area is πab, and the circumference has no closed-form formula but is well-approximated by Ramanujan's π[3(a+b) − √((3a+b)(a+3b))].
This calculator accepts either general-form coefficients or standard-form parameters, computes the center, axes, eccentricity, foci, area, and approximate circumference, and includes presets for common ellipses and a conic-sections reference table.
This center of ellipse calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Input Method, A (x² coefficient), B (xy coefficient), C (y² coefficient) and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Center from general form (B=0): h = −D/(2A), k = −E/(2C) Semi-axes: a = √(rhs/A), b = √(rhs/C) where rhs = −F + D²/(4A) + E²/(4C) Eccentricity: e = c/a, c = √(a²−b²) Area: πab Circumference ≈ π[3(a+b) − √((3a+b)(a+3b))] (Ramanujan)
Result: For method=standard, h=0, k=0, the tool returns the solved center of ellipse outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in center of ellipse formulas and reports derived values, checks, and classifications automatically.
This page is tailored to center of ellipse, with outputs tied directly to the form fields (Input Method, A (x² coefficient), B (xy coefficient), C (y² coefficient)). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
When B=0, complete the square for x and y separately: h = −D/(2A), k = −E/(2C). This gives the center (h, k).
A nonzero B means the ellipse is rotated. You can still find the center, but the axes are tilted. This calculator handles the B=0 case exactly and notes when rotation is present.
Eccentricity e = c/a measures how elongated the ellipse is. For a circle e=0, for a very narrow ellipse e approaches 1.
Compute c = √(a²−b²). The foci are at (h±c, k) if a ≥ b (horizontal major axis) or (h, k±c) if b > a (vertical major axis).
No. The exact circumference involves an elliptic integral. Ramanujan's approximation, π[3(a+b) − √((3a+b)(a+3b))], is highly accurate for most practical cases.
A circle is a special ellipse where a = b (both semi-axes are equal). Its eccentricity is 0, and it has a single center rather than two distinct foci.