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How to use
Use the ellipse calculator to find area, approximate circumference, eccentricity and focal distance from the semi-major and semi-minor axes. Area is exact; circumference uses Ramanujan’s approximation. For full axis lengths of 10 cm and 6 cm, enter semi-axes 5 and 3. Area is 15π ≈ 47.1239 cm², each focus is 4 cm from the center and the two foci are 8 cm apart. Approximate circumference is 25.527 cm. The circle circumference formula applies only when the semi-axes are equal.
- Enter the two semi-axes.
- They may be entered in either order; the tool identifies major and minor automatically.
- Choose the unit.
- Review area, approximate circumference, eccentricity and focal distance.
How it is calculated
Example
For semi-axes 5 and 3: area=15π≈47.124, eccentricity=0.8 and focal distance=4.
Important notes
When the semi-axes are equal, the ellipse is a circle and eccentricity is zero. Circumference is an approximation.
Worked examples and interpreting results
Ellipse formulas use semiaxes, not full axis lengths. If the full axes are 10 and 6, use 5 and 3 in the area formula.
| Case | Calculation | Result |
|---|---|---|
| Area for semiaxes 5 and 3 | π × 5 × 3 | 47.12389 |
| Area in the circular case with radius 3 | π × 3 × 3 | 28.274334 |
| Approximate circumference: semi-axes 5 cm and 3 cm | π × 8 × [1 + 3 × 0.0625 ÷ (10 + √(4 − 3 × 0.0625))] | 25.526999 cm |
| Distance between foci: a=5 cm and b=3 cm | 2 × √(5² − 3²) | 8 cm |
How to check the result
Equal semiaxes give a circle. Ellipse area has a direct formula, while the calculator’s circumference uses an approximation. Do not treat an approximate perimeter as an exact algebraic result for every ellipse.
Ellipse area calculator: semi-major and semi-minor axes
The area of an ellipse is π times its semi-major axis multiplied by its semi-minor axis. If both semi-axes are equal, the shape is a circle. Eccentricity describes how stretched the ellipse is.
- Area with semi-axes 5 and 3: π×5×3 = ≈ 47.1239
- Eccentricity for a=5, b=4: √(1−16÷25) = 0.6
- Equal semi-axes a=b=4: π×4×4 = ≈ 50.2655
Calculation caveat
The inputs are semi-axis lengths, not full diameters. Using full axes directly in the area formula would overstate the area by a factor of four.
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Frequently asked questions
How do I calculate the area of an ellipse?
Use A = πab, where a and b are the semi-major and semi-minor axes.
Is the circumference exact?
Area is exact; circumference uses a highly accurate Ramanujan approximation.
What is eccentricity?
A value from 0 to below 1 describing how stretched the ellipse is.
What is focal distance?
Distance from center to each focus: c=a√(1−b²/a²).
What is the difference between an axis and a semi-axis?
Semi-major a and semi-minor b run from the center to the ellipse edge. The full axis lengths are 2a and 2b; confusing full and half lengths produces incorrect area and perimeter results.
What is the distance between the two foci of an ellipse?
It is 2c = 2√(a² − b²), with a the semi-major axis and b the semi-minor axis. For a=5 and b=3, c=4 and the foci are 8 units apart.