Foci of an Ellipse Calculator — c = √(a²−b²)
Enter the semi-major axis a and semi-minor axis b of an ellipse to find the distance c to each focus (c = √(a²−b²)), the eccentricity, area and perimeter. The diagram shows the ellipse with the two foci marked.
c = √(a² − b²) — foci are at (±c, 0) along the major axis
- 1
Square the semi-major axis (a²)
5² = 25 - 2
Square the semi-minor axis (b²)
3² = 9 - 3
Subtract b² from a²
25 − 9 = 16 - 4
Focal distance c = √(a² − b²)
√16 = 4The two foci are at positions (±c, 0) along the major axis.
How does this calculator work?
For an ellipse with semi-major axis a and semi-minor axis b, the focal distance is c = √(a²−b²) and the two foci are at positions (±c, 0) on the major axis. Eccentricity e = c/a (0 = circle, near 1 = very elongated). Area = π·a·b.
Formula
How this is calculated
An ellipse is the set of all points where the sum of distances to two fixed points (the foci) is constant. The foci lie on the major axis at a distance c from the centre, where c = √(a² − b²), a is the semi-major axis (the longer half-radius) and b is the semi-minor axis (the shorter half-radius). If a = b the ellipse degenerates into a circle and c = 0 (foci merge at the centre).
Eccentricity e = c/a measures how "squashed" the ellipse is: e = 0 is a perfect circle; e approaching 1 is a very elongated ellipse (at e = 1 it becomes a parabola). Area is the exact formula π·a·b. The perimeter has no simple closed form; this calculator uses Ramanujan's second approximation π(a+b)(1 + 3h/(10+√(4−3h))) where h = (a−b)²/(a+b)², accurate to within 0.03% for any eccentricity.
Ellipses appear throughout science: planetary orbits (Kepler's first law), satellite paths, reflective properties of elliptical mirrors, and whispering galleries.
Frequently asked questions
The Pythagorean-style identity a² = b² + c² always holds. The semi-major axis a is the hypotenuse of a right triangle whose legs are b (semi-minor) and c (focal distance). Knowing any two of a, b, c lets you find the third.
Eccentricity e = c/a is a dimensionless shape parameter: 0 means a perfect circle, values near 1 mean a very elongated ellipse. The Earth's orbit has e ≈ 0.017 (nearly circular); Halley's Comet has e ≈ 0.967 (very elongated). An eccentricity ≥ 1 is no longer an ellipse (it becomes a parabola at exactly 1, or a hyperbola above 1).
Unlike a circle, the perimeter of an ellipse cannot be expressed with elementary functions. The exact formula involves an elliptic integral. Ramanujan's second approximation (used here) is accurate to within 0.03% for any eccentricity, which is sufficient for almost all practical purposes.
Also known as
TG we-Calculate Editorial Team. (2026). Foci of an Ellipse Calculator — c = √(a²−b²) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/foci-of-ellipse-calculator
TG we-Calculate Editorial Team. "Foci of an Ellipse Calculator — c = √(a²−b²)." TG we-Calculate. 2026. https://we-calculate.com/calculator/foci-of-ellipse-calculator.
TG we-Calculate Editorial Team, "Foci of an Ellipse Calculator — c = √(a²−b²)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/foci-of-ellipse-calculator
@misc{wecalculate_foci_of_ellipse_calculator, title = {Foci of an Ellipse Calculator — c = √(a²−b²)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/foci-of-ellipse-calculator}}, year = {2026}, note = {TG we-Calculate} }
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