Circangle

The ellipse calculator

Give the two semi-axes of any ellipse and get its exact circumference and area. The circumference has no exact formula in terms of π; it needs the complete elliptic integral E(k). It opens on G5, a near-golden ellipse from the book.

Decimals or fractions such as 99/17. Any unit; the circumference comes out in the same unit and the area in its square.
semi-axis asemi-axis barc, C/4foci
Circumference
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Area
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How the π shortcuts do on this ellipse
MethodCircumferenceError

The formulas

Take a as the longer semi-axis. The shape is carried by the modulus k, with , so k = 0 is the circle and k → 1 flattens the ellipse to a line.

The circumference is four circangle arcs, , where E(k) is the complete elliptic integral of the second kind. The area is . Here K(0) is the quarter turn, the value of the complete elliptic integral of the first kind for the circle; numerically it is π/2, so 2K(0)ab is the familiar πab.

For a circle, E(0) = K(0), and the circumference becomes 4K(0)r, the circangle form of 2πr. This calculator evaluates K and E with the arithmetic–geometric mean, to about twelve significant figures.