# Ellipse Calculator

Calculate ellipse area, circumference, eccentricity, and foci from the semi-major and semi-minor axes. Free online ellipse calculator with Ramanujan's.

## What this calculates

Calculate the area, approximate circumference, eccentricity, and focal points of any ellipse. Enter the semi-major and semi-minor axes for instant, accurate results using Ramanujan's approximation.

## Inputs

- **Semi-Major Axis (a)** — The longer radius of the ellipse.
- **Semi-Minor Axis (b)** — The shorter radius of the ellipse.

## Outputs

- **Area** — The area of the ellipse (pi x a x b).
- **Circumference (approx.)** — Approximate perimeter using Ramanujan's formula.
- **Eccentricity** — How elongated the ellipse is (0 = circle, approaching 1 = very elongated).
- **Focal Distance (c)** — The distance from center to each focus.
- **Foci Positions** — formatted as text — The coordinates of the two foci (assuming center at origin).

## Details

An ellipse is a closed curve where the sum of distances from any point to two fixed points (foci) is constant. A circle is a special case of an ellipse where both axes are equal.

Ellipse Formulas

- Area: A = pi x a x b, where a is the semi-major axis and b is the semi-minor axis.

- Circumference: There is no exact closed-form formula. Ramanujan's approximation is: C ≈ pi(a+b)(1 + 3h/(10 + sqrt(4 - 3h))), where h = ((a-b)/(a+b))^2.

- Eccentricity: e = c/a, where c = sqrt(a^2 - b^2). For a circle, e = 0. As the ellipse elongates, e approaches 1.

- Foci: Located at distance c from the center along the major axis.

Ellipses are fundamental in astronomy (planetary orbits are ellipses with the Sun at one focus, per Kepler's first law), optics (elliptical reflectors), architecture (whispering galleries), and engineering.

## Frequently Asked Questions

**Q: How do I calculate the area of an ellipse?**

A: The area of an ellipse is A = pi x a x b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). For an ellipse with a = 5 and b = 3: A = pi x 5 x 3 = 47.12 square units. This generalizes the circle area formula (where a = b = r).

**Q: Why is there no exact formula for the circumference of an ellipse?**

A: Unlike a circle, the circumference of an ellipse cannot be expressed with elementary functions. It requires an elliptic integral, which is an infinite series. Ramanujan's approximation is extremely accurate for most practical purposes, with error less than 0.001% for eccentricities below 0.95.

**Q: What is eccentricity?**

A: Eccentricity measures how much an ellipse deviates from being a circle. It ranges from 0 (a perfect circle) to values approaching 1 (very elongated). Earth's orbit has eccentricity 0.0167 (nearly circular), while Halley's Comet has eccentricity 0.967 (very elongated). It is calculated as e = c/a where c is the focal distance.

**Q: What are the foci of an ellipse?**

A: The foci (singular: focus) are two special points inside the ellipse. The defining property of an ellipse is that for any point on the curve, the sum of distances to the two foci is constant (equal to 2a). The foci are located at distance c = sqrt(a^2 - b^2) from the center along the major axis.

**Q: How are ellipses related to planetary orbits?**

A: Kepler's first law states that planets orbit the Sun in ellipses with the Sun at one focus. This was a revolutionary discovery, replacing the ancient belief in perfect circular orbits. The eccentricity of a planet's orbit determines how much its distance from the Sun varies throughout the year.

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Source: https://vastcalc.com/calculators/math/ellipse
Category: Math
Last updated: 2026-04-21