Softonic

Ellipse Calculator

Step 1 / 3 Area
Next

An ellipse is the oval you get by stretching a circle along one direction. It is fully described by two numbers: the semi-major axis (a), the longest half-width, and the semi-minor axis (b), the shortest. Enter those two values and this calculator returns the area, the perimeter (using the accurate Ramanujan approximation), the focal distance c, and the eccentricity that measures how stretched the oval is. It is handy for orbits, optics, elliptical tracks, garden beds, machine parts, or any geometry question where πab is faster done for you than by hand.

How the ellipse values are calculated

  1. 1

    Enter the semi-major axis (a)

    Half the length of the longest diameter: the distance from the centre to the farthest point on the curve. Any unit works; the area returns in that unit squared and the lengths in that same unit.

  2. 2

    Enter the semi-minor axis (b)

    Half the length of the shortest diameter. If a and b are equal the ellipse is simply a circle, and the eccentricity comes back as zero.

  3. 3

    Read the results

    Area = πab, perimeter uses the Ramanujan approximation, focal distance c = √(a² − b²) and eccentricity e = c / a. Everything updates instantly as you type.

The formulas

An ellipse with semi-major axis a (the larger half-axis) and semi-minor axis b (the smaller) has these properties:

  • Area: A = π · a · b
  • Perimeter (Ramanujan’s first approximation): P ≈ π · [ 3(a + b) − √((3a + b)(a + 3b)) ]
  • Focal distance: c = √(a² − b²), the distance from the centre to each focus
  • Eccentricity: e = c / a, a value between 0 (a circle) and just under 1 (a very flattened oval)

There is no simple exact formula for an ellipse’s perimeter, it is an elliptic integral, so this tool uses Ramanujan’s famous approximation, which is accurate to better than 0.04% even for fairly stretched shapes.

Worked example

Take a semi-major axis of 5 and a semi-minor axis of 3 (so a = 5, b = 3).

  • Area = π × 5 × 3 = 47.1239
  • Perimeter ≈ π × [ 3(5 + 3) − √((15 + 3)(5 + 9)) ] = π × [ 24 − √252 ] ≈ 25.5270
  • Focal distance c = √(5² − 3²) = √16 = 4.0000
  • Eccentricity e = 4 / 5 = 0.8000

Reference values

a b Area Perimeter Focal c Eccentricity
5 5 78.5398 31.4159 0.0000 0.0000
5 3 47.1239 25.5270 4.0000 0.8000
6 4 75.3982 31.7309 4.4721 0.7454
10 2 62.8319 42.0112 9.7980 0.9798

Pitfalls to avoid

  • Semi-axes, not full axes. Use half the diameter for each axis. Entering the full width and height gives an area four times too big.
  • a must be the larger one. The focal distance and eccentricity are defined from the longer axis; this tool sorts a and b for you, so order does not matter.
  • A circle is a special case. When a = b the focal distance and eccentricity are both 0, that is correct, not an error.
  • Keep units consistent. Both axes in centimetres gives area in cm² and all lengths in centimetres. Do not mix units.

Frequently Asked Questions

The semi-major axis (a) is half of the ellipse’s longest diameter; the semi-minor axis (b) is half of its shortest. If you measured the full width and height of the oval, divide each by two before entering them here.

Unlike the area, an ellipse’s exact perimeter cannot be written with elementary functions, it is an elliptic integral. This calculator uses Ramanujan’s first approximation, which is accurate to better than 0.04% for typical shapes and far closer than the naive π(a + b) estimate.

Eccentricity measures how stretched the ellipse is, on a scale from 0 to almost 1. An eccentricity of 0 is a perfect circle; values near 1 describe a long, thin, cigar-shaped oval. It equals c / a, the focal distance divided by the semi-major axis.

No. The numbers you enter are sent to our server only to run the calculation, and they are not stored, saved, or shared with anyone. On the guided multi-step pages, the two values also travel in the page link so the next step can continue where you left off.

Related Tools

BMI Calculator

Calculate body mass index from height and weight. Shows WHO category, healthy-weight range and limitations of BMI.

Social Security Calculator

Estimate a US Social Security retirement benefit from your PIA and claiming age. Compare monthly and annual amounts at 62, FRA 67 and 70.

Age Calculator

Calculate exact age in years, months and days from a birth date, plus total days, hours and the next birthday countdown.

Dog Age Calculator

Convert your dog's age to human years using modern veterinary research. Breed-size aware: small, medium and large dogs age differently.

Amp-Hour Calculator

Work out battery runtime and stored energy from amp-hours, voltage and load current. Get runtime in hours and minutes plus energy in watt-hours and kilowatt-hours.

Fahrenheit to Celsius Converter

Convert Fahrenheit temperatures to Celsius with the exact formula, Kelvin output and common reference points for weather, cooking and health.

Tool available in other languages