Understand the ellipse

An ellipse is the set of points whose distances to two fixed foci have a constant sum. Its semi-major axis a is at least its semi-minor axis b. Full axis lengths are 2a and 2b.

Properties that make the calculations work

Area is exactly πab in the mathematical model. Eccentricity is √(1−b²/a²), and ranges from zero for a circle to values approaching one for very elongated ellipses. The perimeter does not have a simple elementary formula.

Where the model is useful

Use the semi-axis or full-axis calculator according to how your drawing is labelled. Perimeters here use Ramanujan’s second approximation and are explicitly marked approximate; precision-sensitive manufacturing needs an appropriate exact numerical method.

Measure the right thing

Semi-axes are half the full axis lengths. The approximation label applies to perimeter, not to the area formula.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k². This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Ellipse provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.