Center-to-focus distance and interpretation
Focus distance is measured from the center to either focus; the two foci are twice that distance apart. Eccentricity is dimensionless. The perimeter is an approximation rather than an exact elementary formula.
To calculate area from the dimensions of a supported plane shape, use the Area Calculator and follow its own input conventions.
The formula
Use the difference of the squared semi-axes for focal distance and divide it by the major semi-axis for eccentricity. Area is πab; perimeter uses Ramanujan’s second approximation.
Verified worked example
An ellipse with semi-axes 5 and 3 has center-to-focus distance √(25 − 9) = 4 and eccentricity 0.8. Its area is 15π, approximately 47.123890 square units.
How to use this calculator
- Enter the major semi-axis, equal to half of the longer full axis.
- Enter the minor semi-axis, equal to half of the shorter full axis.
- Read focus distance and eccentricity; treat the displayed perimeter as an approximation.
Ellipse input conventions
A semi-axis measures from the center to the outline, so it is half of the full axis length. The major semi-axis must be at least as large as the minor one. The two foci lie symmetrically on the major axis, each at the reported distance from the center. Their separation is twice that distance. Equal semi-axes produce a circle with coincident foci and zero eccentricity.
Interpreting center-to-focus distance
Area follows directly from the semi-axis product, but ellipse perimeter has no comparable elementary exact formula. The displayed perimeter uses Ramanujan’s second approximation with h = ((a − b)/(a + b)) squared. It is an estimate rather than a numerical elliptic-integral evaluation. Near extremely elongated shapes, verify perimeter against a higher-precision method if small error materially affects a cutting or manufacturing specification.
Assumptions & limitations
What this calculation assumes
- Semi-axes are positive and the major semi-axis is at least the minor semi-axis.
What to keep in mind
- Axis lengths only; perimeter is an approximation and rotated coordinate equations are not generated.
Common questions
Should I enter the full width and height?
No. Divide each full axis by two before entering it. For an outline 10 units wide and 6 units tall, enter semi-axes 5 and 3.
Where are the foci?
For a horizontal major axis centered at the origin, they are at (−c, 0) and (c, 0). For a vertical major axis, exchange the coordinates.
What happens when both semi-axes are equal?
The ellipse is a circle. Focus distance and eccentricity become zero, and the perimeter approximation reduces to the exact circle circumference 2πa.
Sources & further reading
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