Free · Mathematical relations and geometry

Ellipse Semiaxes, Area and Approximate Perimeter

Estimate ellipse perimeter from two semiaxes and calculate its exact area and eccentricity.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

Semiaxes 3 and 2 give area 6π and approximate perimeter 15.865440. Eccentricity is √5/3, approximately 0.745356.

A=πab; P≈π(a+b)[1+3h/(10+√(4−3h))], h=((a−b)/(a+b))²

Worked example

Enter these known values and leave the other values blank.

Semi-major axis (a)
3
Semi-minor axis (b)
2
Perimeter
15.865
Area
18.85
Eccentricity
0.7454

Assumptions and limitations

  • a and b are positive semiaxes, half the full axis lengths. Inputs share a chosen length unit; the interface has no unit conversion controls.
  • The perimeter uses Ramanujan’s second approximation. It is not the exact elliptic-integral perimeter, especially near a collapsed ellipse.
  • Eccentricity is dimensionless and uses the larger semiaxis as major axis.
  • Area is exact for the model. A known area and one semiaxis can recover the other; perimeter is forward only.

Common questions

How can I solve for a missing quantity?

Leave the unknown control empty and enter the quantities needed by the selected relation. Read the assumptions for any branch or domain restrictions.

How should I interpret the worked example?

It is an exact mathematical model with independently checked numbers. Display rounding does not change the values used for subsequent calculations.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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