Ellipse Standard Form Calculator — Center, Foci & Eccentricity
Enter the center (h, k) and semi-axes a (x-direction) and b (y-direction) to get the standard equation, the locations of the two foci, eccentricity, area, perimeter, and a scaled plot of the ellipse.
- 1
Focal distance c
√(5² − 3²) = 4 - 2
Eccentricity e
4 ÷ 5 = 0.8000e = 0 is a perfect circle; e approaching 1 is a very flat ellipse.
How does this calculator work?
The ellipse (x−h)²/a² + (y−k)²/b² = 1 has center (h,k), semi-axes a and b, and foci at distance c = √|a²−b²| from the center along the major axis. Eccentricity e = c/max(a,b) — 0 for a circle, approaching 1 for a very flat ellipse.
Formula
How this is calculated
An ellipse in standard position is fully described by its center (h, k) and its two semi-axes: a along the x-direction and b along the y-direction. The standard form equation is (x−h)²/a² + (y−k)²/b² = 1. If a > b the major axis is horizontal; if b > a it is vertical.
The focal distance c = √(a²−b²) (using the larger minus the smaller) locates the two foci along the major axis at distance c from the center. Eccentricity e = c/max(a,b) measures elongation: e = 0 is a perfect circle, and e approaching 1 is a very flat ellipse. Every point on the ellipse has the property that the sum of its distances to the two foci equals 2 × max(a, b).
Area = π × a × b (exact). Perimeter uses Ramanujan's second approximation, accurate to within 0.02% across all eccentricities. The plot draws the full ellipse curve with center and foci marked.
Frequently asked questions
The standard form (x−h)²/a² + (y−k)²/b² = 1 tells you the center is (h, k) and the semi-axes are a (horizontal) and b (vertical). The larger denominator corresponds to the major axis. Expand the squares to convert to general conic form Ax²+Cy²+Dx+Ey+F=0 if needed.
Calculate c = √(major² − minor²) where major = max(a, b) and minor = min(a, b). If the major axis is horizontal, the foci are at (h ± c, k). If the major axis is vertical, they are at (h, k ± c). The foci always lie on the major axis, inside the ellipse.
Eccentricity e = c/major ranges from 0 to 1 for an ellipse. At e = 0 the shape is a perfect circle. As e approaches 1 the ellipse becomes increasingly elongated (the minor axis shrinks relative to the major). Earth's orbit has e ≈ 0.017 (nearly circular); a very eccentric comet orbit might have e ≈ 0.97.
Also known as
TG we-Calculate Editorial Team. (2026). Ellipse Standard Form Calculator — Center, Foci & Eccentricity [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/ellipse-standard-form-calculator
TG we-Calculate Editorial Team. "Ellipse Standard Form Calculator — Center, Foci & Eccentricity." TG we-Calculate. 2026. https://we-calculate.com/calculator/ellipse-standard-form-calculator.
TG we-Calculate Editorial Team, "Ellipse Standard Form Calculator — Center, Foci & Eccentricity," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/ellipse-standard-form-calculator
@misc{wecalculate_ellipse_standard_form_calculator, title = {Ellipse Standard Form Calculator — Center, Foci & Eccentricity}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/ellipse-standard-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
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