Find the foci of an ellipse from semi-axes or the general equation. Computes foci coordinates, distance between foci, eccentricity, linear eccentricity, vertices, center, and key ellipse properties.
The foci (singular: focus) are two special points inside an ellipse with a remarkable property: for every point on the ellipse, the sum of the distances to the two foci is constant and equal to the length of the major axis (2a). This defining property is what makes an ellipse an ellipse.
The foci lie along the major axis, each at a distance c from the center, where c = √(a² − b²) and a and b are the semi-major and semi-minor axes respectively. The quantity c is called the linear eccentricity, and the ratio e = c/a is the eccentricity, which ranges from 0 (a circle, where the foci coincide at the center) to just below 1 (a very elongated ellipse where the foci approach the vertices).
The position of the foci has deep significance in physics and engineering. Kepler's first law states that planets orbit the Sun in ellipses with the Sun at one focus. Elliptical rooms and "whispering galleries" exploit the reflection property: sound emitted from one focus converges at the other. Satellite dish receivers and medical lithotripters use ellipsoidal reflectors based on the same principle.
This calculator accepts either the semi-axes (a, b) with an optional center offset, or the coefficients of the general equation Ax²+Cy²+Dx+Ey+F = 0, and computes the foci coordinates, distance between foci, eccentricity, linear eccentricity, vertices, co-vertices, area, and the semi-latus rectum. Presets cover common ellipses and a key-points table lists all important coordinates.
The Foci of an Ellipse Calculator — From Semi-axes or Equation is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Focus 1, Focus 2, Distance Between Foci 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.
Linear eccentricity: c = √(a² − b²) where a ≥ b Foci (horizontal major axis): (h ± c, k) Foci (vertical major axis): (h, k ± c) Eccentricity: e = c/a Distance between foci: 2c Constant sum property: d₁ + d₂ = 2a for any point on ellipse Semi-latus rectum: ℓ = b²/a
Result: Foci at (−4, 0) and (4, 0), c = 4, e = 0.8
c = √(25 − 9) = √16 = 4. Since a > b, the major axis is horizontal. Foci: (0 ± 4, 0) = (−4, 0) and (4, 0). Eccentricity e = 4/5 = 0.8. Distance between foci = 2 × 4 = 8.
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.
Foci of an Ellipse Calculator — From Semi-axes or Equation 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.
Compute c = √(a² − b²) where a is the semi-major axis and b is the semi-minor axis. The foci are at (h ± c, k) for a horizontal ellipse or (h, k ± c) for a vertical ellipse.
c (linear eccentricity) is the actual distance from the center to each focus, measured in the same units as the axes. e (eccentricity) is the dimensionless ratio c/a, always between 0 and 1 for an ellipse.
When c = 0 (i.e., a = b), the two foci merge at the center and the ellipse becomes a circle. Every point on a circle is equidistant from the center.
An ellipse is the set of all points where the sum of distances to the two foci equals 2a (the major axis length). This is the string-and-pins construction method.
The semi-latus rectum ℓ = b²/a is the distance from a focus to the ellipse along a line perpendicular to the major axis through that focus. It appears in the polar form r = ℓ/(1 − e cos θ).
Yes. First convert to standard form by completing the square to find h, k, a, b. Then c = √(a² − b²) and the foci follow from the center and major axis direction.