Ellipse Calculator

An ellipse is the set of points whose combined distance from two fixed focal points is constant. It resembles a stretched or compressed circle and is described most conveniently by two perpendicular semi-axes. The semi-major axis, conventionally called a, is the longer radius from the center to the curve. The semi-minor axis, called b, is the shorter radius. Entering these two lengths is enough to derive the ellipse area, approximate perimeter, eccentricity, and distance from the center to either focus. The area formula is exact and multiplies pi by the two semi-axis lengths. This reflects how scaling a unit circle horizontally by a and vertically by b scales its area by their product. The perimeter is subtler because an ellipse has no elementary closed-form circumference formula. This calculator uses Ramanujan's second approximation, which is extremely accurate across ordinary ellipse shapes. It evaluates a dimensionless shape factor based on the squared difference and sum of the axes, then applies a correction to pi times their sum. Eccentricity measures how far an ellipse departs from a circle. The focal distance is the square root of the difference between the squared semi-major and semi-minor axes, and eccentricity divides that distance by the semi-major axis. A circle has equal axes, zero focal distance, and zero eccentricity. As an ellipse becomes increasingly narrow, its eccentricity approaches one without reaching it for a nondegenerate ellipse. Each focus lies along the major axis at the calculated focal distance from the center, so the separation between the two foci is twice that value. Ellipse calculations occur in coordinate geometry, astronomy, architecture, optics, acoustics, mechanical design, and orbital models. All input lengths must use the same unit. Area is then expressed in the square of that unit, perimeter and focal distance remain in the original unit, and eccentricity has no unit. This calculator assumes a is at least b; if your measurements are reversed, swap them before calculating. Displayed decimals are rounded to six places, so retain additional precision when carrying a result into engineering work. A quick check is that the area must be positive, eccentricity must fall from zero up to but not including one, and the focal distance cannot exceed the semi-major axis.