How this calculation works
Treat the larger semiaxis as a and the smaller as b. Compute area, full axes and focal geometry, while labeling Ramanujan’s second perimeter approximation separately from an exact elliptic integral.
Area = πab; approximate perimeter = π(a+b)[1 + 3h/(10+√(4−3h))], h=((a−b)/(a+b))²A worked example
Semiaxes 5 m and 3 m give area 15π ≈ 47.123890 m². Full axes are 10 m and 6 m; each focus is 4 m from the center. The approximate perimeter is about 25.526999 m.
Good to know
A nondegenerate planar ellipse with positive semiaxes. Perimeter uses Ramanujan II, not exact quadrature, and is less accurate for very thin ellipses. No tilted coordinate fit or material allowance.
Reference: MathWorld: geometry definitions and formulas (opens in a new tab) ↗
A semiaxis starts at the center
The default center-to-edge lengths 5 m and 3 m describe full widths of 10 m and 6 m. Entering those full widths as semiaxes would multiply the area by four. The calculator sorts the two positive inputs so their order does not change the geometry.
Area is π × 5 × 3 = 15π, about 47.123890 m². The center-to-focus distance is √(5²−3²) = 4 m, so the foci are 8 m apart. The focus is not the same point as the endpoint of a semiaxis.
The circle calculator supplies the equal-semiaxis case. An ellipse perimeter here is an approximation, while its πab area uses the stated ellipse model.
Area and perimeter do not have the same formula accuracy
Ellipse area has the direct πab relationship. Its exact perimeter is described by an elliptic integral rather than a simple circle-style circumference expression. This tool uses Ramanujan’s second perimeter approximation and labels it accordingly.
For 5 m and 3 m semiaxes, that approximation gives about 25.526999 m. It reduces to the circle expression when both semiaxes are equal. A very thin ellipse is a different numerical case; the approximation is not presented as an exact quadrature or an unlimited-precision measured length.
Scale both lengths to compare the same shape at another size
Doubling both default semiaxes gives four times the area and twice the perimeter approximation. Changing only one semiaxis changes the shape as well as its size. The comparison table scales both together so those effects are not mixed.
The circle calculator covers the equal-semiaxis case. The regular polygon calculator uses a different boundary made of equal straight sides. Keep every input in one length unit; the unit selector labels the result and does not silently convert numbers already entered.
A few questions, answered
Are the input lengths full widths?
No. They are center-to-edge semiaxes. The full widths are twice those inputs.
Is the perimeter exact?
No. It uses the explicitly named Ramanujan II approximation, not an exact elliptic-integral evaluation.
Can the first input be smaller?
Yes. The larger becomes the semimajor axis and the smaller the semiminor axis.
Do equal semiaxes give a circle?
Yes. Focal distance becomes zero and the area/perimeter relationships reduce to the circle case.