Ellipse
Calculate all properties of an ellipse given semi-major axis (a) and semi-minor axis (b). Free online ellipse calculator computes area, perimeter using Ramanujan approximation, eccentricity, and linear eccentricity.
About This Calculator
The Ellipse Calculator computes all key properties of an ellipse from its semi-major axis (a) and semi-minor axis (b). An ellipse is a conic section defined as the set of points where the sum of distances to two fixed foci is constant -- it is essentially a stretched or compressed circle.
Key formulas: Area = piab. Circumference (Ramanujan) ≈ pi(a+b)(1 + 3h/(10 + sqrt(4-3h))) where h = ((a-b)/(a+b))^2. Eccentricity e = sqrt(1 - b^2/a^2). Linear eccentricity c = ae = sqrt(a^2 - b^2).
Ellipses are fundamental in astronomy (Kepler's laws of planetary motion describe elliptical orbits), physics (harmonic oscillators), engineering (gear design), and architecture (elliptical arches and domes). Understanding ellipse properties is essential for geometry, precalculus, and calculus students.
Frequently Asked Questions
What is an ellipse?
An ellipse is a curved shape defined as the set of all points where the sum of distances to two fixed points (foci) is constant. It looks like a stretched circle and has two axes: the longer semi-major axis (a) and the shorter semi-minor axis (b).
What is the area of an ellipse?
The area of an ellipse is calculated using the formula A = pi x a x b, where a is the semi-major axis and b is the semi-minor axis. For example, an ellipse with a = 5 and b = 3 has an area of approximately 47.12 square units.
How is ellipse circumference calculated?
The ellipse circumference (perimeter) uses the Ramanujan approximation: C ≈ pi(a + b)(1 + 3h/(10 + sqrt(4 - 3h))) where h = ((a - b)/(a + b))^2. This formula provides accuracy within about 0.04% for most ellipses.
What is ellipse eccentricity?
Eccentricity (e) measures how elongated an ellipse is. It ranges from 0 (perfect circle) to 1 (highly elongated). Calculated as e = sqrt(1 - b^2/a^2) where a is the semi-major axis and b is the semi-minor axis.
What are the real-world applications of ellipses?
Ellipses appear in planetary orbits (Kepler's laws), satellite dish design (parabolic reflectors), medical imaging (CT scans), architecture (elliptical arches), and engineering (cam gears and elliptical trainers).
Is the Ellipse calculator free to use?
Yes, all calculators on Calculy including this ellipse calculator are completely free to use with no registration or limits.