Convert an ellipse between standard form and general form. Find center, semi-axes, eccentricity, foci, vertices, co-vertices, area, and circumference from either equation form.
An ellipse can be described in two main algebraic forms: standard form and general form. The standard form, (x−h)²/a² + (y−k)²/b² = 1, immediately reveals the center (h, k) and the semi-axes a and b. The general form, Ax² + Bxy + Cy² + Dx + Ey + F = 0, is more compact but hides these geometric properties behind its coefficients.
Converting between the two forms is a fundamental skill in analytic geometry. To go from general to standard form, you complete the square for both x and y terms, then divide to get 1 on the right side. To go from standard to general form, you expand the squared terms, multiply through by a²b², and rearrange.
Once in standard form, extracting the ellipse's properties is straightforward. The eccentricity e = √(1 − b²/a²) for a ≥ b tells you the shape — 0 for a circle, approaching 1 for a very elongated ellipse. The foci lie at distance c = √(a²−b²) from the center along the major axis. Vertices sit at the ends of the major axis and co-vertices at the ends of the minor axis.
This calculator accepts either standard form parameters (center and semi-axes) or general form coefficients (A through F), performs the conversion, and displays all key properties including center, semi-axes, eccentricity, foci, vertices, co-vertices, area, and approximate circumference. Presets for common ellipses and a reference table for ellipse forms are included.
The Ellipse Standard Form Calculator — Convert Between Forms is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Standard Form, Center, Semi-axis a in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Standard form: (x−h)²/a² + (y−k)²/b² = 1 General form: Ax²+Bxy+Cy²+Dx+Ey+F = 0 General → Standard: h = −D/(2A), k = −E/(2C), rhs = −F + D²/(4A) + E²/(4C) a = √(rhs/A), b = √(rhs/C) Eccentricity: e = √(1 − min²/max²) Foci distance: c = √(a² − b²) from center along major axis
Result: Standard form: (x−2)²/9 + (y+3)²/4 = 1, center = (2, −3)
Complete the square: 4(x²−4x) + 9(y²+6y) = −61 → 4(x−2)²−16 + 9(y+3)²−81 = −61 → 4(x−2)² + 9(y+3)² = 36 → (x−2)²/9 + (y+3)²/4 = 1. Center is (2, −3), a = 3, b = 2.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Ellipse Standard Form Calculator — Convert Between Forms calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
Group x and y terms separately, complete the square for each, then divide by the constant on the right side to get 1. This reveals center (h, k) and semi-axes a, b.
A nonzero B (the xy coefficient) means the ellipse is rotated. You need to rotate coordinates to eliminate the xy term first. This calculator handles axis-aligned ellipses (B = 0).
Vertices are the endpoints of the major (longer) axis. Co-vertices are the endpoints of the minor (shorter) axis. They are perpendicular to each other at the center.
The major axis corresponds to the larger denominator in standard form. If a > b, the major axis is horizontal; if b > a, it is vertical.
Yes: x = h + a cos(t), y = k + b sin(t) for t ∈ [0, 2π). This traces out the ellipse and is especially useful for plotting or animations.
For Ax²+Cy²+Dx+Ey+F = 0 (B=0): if A and C are both positive (or both negative), it's an ellipse. If A = C, it's a circle. If A and C have different signs, it's a hyperbola.