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Ellipse Calculator: Area, Perimeter and Eccentricity

This free ellipse calculator finds the area, the perimeter, the eccentricity and the focal distance of an ellipse from its two semi-axes. The perimeter uses Ramanujan’s accurate approximation.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

An ellipse is a stretched circle. Planets orbit in ellipses, and ovals appear in tables, tracks, mirrors and logos.

Enter the two semi-axes.

How to Use This Tool

  1. Enter the value or values asked for in the boxes.
  2. Choose the unit label if there is a choice.
  3. Press Calculate.
  4. Read every result in the table and copy what you need.

How the Calculation Works

Area = πab. Perimeter ≈ π(3(a + b) − √((3a + b)(a + 3b))) (Ramanujan). Focal distance c = √(a² − b²) and eccentricity e = c ÷ a. If you enter b longer than a, the tool swaps them.

Worked Example

For a = 5 and b = 3, the area is 47.1239, the perimeter about 25.5254, the focal distance 4 and the eccentricity 0.8.

Common Uses

  • Running tracks and ovals.
  • Orbits in astronomy.
  • Mirrors and lenses.
  • Design and tables.

Tips and Common Mistakes

  • Semi-axes are half the full width and length.
  • Equal axes give a circle.
  • Use the same unit for both.
  • Measure carefully: a small error in a length is squared in an area and cubed in a volume, so a 5 percent error in a side can become a 10 percent error in the area or a 15 percent error in the volume.
  • Round only at the end. If you copy a result into another calculation, copy the full number from the tool rather than a rounded one, so small errors do not build up.

Next, you may need the circle calculator, arc length calculator, trapezoid calculator, rectangle calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

Area = π × a × b, where a and b are the semi-axes.
There is no simple exact formula. Ramanujan’s formula is accurate to a tiny fraction of a percent for most shapes.
A measure of how stretched the ellipse is: 0 is a circle and values near 1 are very long and thin.
Two points inside the ellipse such that the sum of the distances from any point on the edge to them is constant. They lie at a distance c = √(a² − b²) from the centre.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.