What formula does this calculator use?

It uses Ramanujan's first approximation: P ≈ π × [3(a+b) − √((3a+b)(a+3b))], where a is the semi-major axis and b is the semi-minor axis. This formula is accurate to within about 0.1% for most ellipses, with the error rising slightly for highly eccentric shapes.

Why is there no exact formula for ellipse perimeter?

The exact perimeter requires an elliptic integral, which has no closed form in terms of elementary functions. Ramanujan's approximation is the most practical and accurate alternative for everyday use.

What is the semi-major axis?

The semi-major axis (a) is half the longest diameter of the ellipse. The semi-minor axis (b) is half the shortest diameter. Both must be positive and a must be greater than or equal to b.

What happens when a equals b?

When a = b the ellipse becomes a circle. Ramanujan's formula correctly reduces to P = 2πa, matching the standard circle circumference formula.

What is eccentricity and how is it calculated?

Eccentricity (e) measures how stretched the ellipse is: e = √(1 − b²/a²). A value of 0 means a perfect circle; a value approaching 1 means a very elongated ellipse.

Is this calculation done on my device?

Yes. All computations run entirely in your browser using JavaScript. No data is sent to any server.

Perimeter of an Ellipse Calculator

Name: Perimeter of an Ellipse Calculator
Availability: InStock
Author: Nham Vu

Enter the semi-major and semi-minor axes of an ellipse to calculate its perimeter using Ramanujan's highly accurate approximation.

Ellipse Dimensions

Unit

Semi-major axis (a — longest radius)

Semi-minor axis (b — shortest radius)

Result

Enter the semi-major and semi-minor axes to see the perimeter.

Perimeter (Circumference)

Diagram

Property	Value
Semi-major axis (a)
Semi-minor axis (b)
Eccentricity (e)
Area
Perimeter

Summary

Enter the semi-major and semi-minor axes of an ellipse to calculate its perimeter using Ramanujan's highly accurate approximation.

How it works

Enter the semi-major axis (a) — the longest radius of the ellipse.
Enter the semi-minor axis (b) — the shortest radius of the ellipse.
Choose your preferred unit of measurement.
The tool applies Ramanujan's formula: P ≈ π × [3(a+b) − √((3a+b)(a+3b))].
The perimeter, eccentricity, and area are displayed instantly.
Copy the result with one click.

Use cases

Calculate the length of an oval running track or garden path.
Determine how much trim or edging is needed for an elliptical table.
Estimate ribbon or framing material for oval picture frames.
Solve geometry and calculus homework problems involving ellipses.
Compute orbital path length approximations in basic astronomy.
Find the border length of an elliptical pool or fountain.
Verify CAD measurements for elliptical architectural features.
Cross-check ellipse perimeter estimates in engineering drafts.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu