Ellipse Calculator

An ellipse is the smooth oval you get when a circle is stretched along one direction, the shape of planetary orbits, whispering galleries, running tracks and countless engineering and design forms, and this calculator works out all of its key properties from just the two semi-axes. Enter the semi-major axis, the distance from the centre to the far edge, and the semi-minor axis, the distance to the nearer edge, and it returns the area, the perimeter, the eccentricity, the distance to the foci and the full major and minor axis lengths, updating as you type. The area is wonderfully simple, just pi times the two semi-axes multiplied together, a neat generalisation of the circle's pi r squared that the circle becomes when the two axes are equal. The perimeter, by contrast, is famously awkward: there is no simple exact formula, so this calculator uses Ramanujan's celebrated second approximation, which is accurate to a tiny fraction of a percent for all but the most extremely stretched ellipses, far better than the rough estimates many tools use. It also reports the eccentricity, the number from 0 to 1 that captures how elongated the ellipse is, and the foci, the two special interior points that define the ellipse and matter so much in astronomy and optics. That makes the tool genuinely useful for students meeting conic sections, for engineers and designers laying out elliptical arches, tanks, gears and tracks, for anyone working out the material around an oval, and for understanding orbits in physics. Because every value updates live, you can watch a near-circle stretch into a long thin ellipse and see the eccentricity climb. The formulas and a worked example are explained clearly below.

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