Everyday utility

Ellipse Calculator

Enter the semi-major axis (a) and semi-minor axis (b) to calculate the area, perimeter, and eccentricity of an ellipse. The perimeter uses Ramanujan's accurate approximation — there is no exact closed-form formula.

Last reviewed May 19, 2026 by ToolSpilo Editorial Team.

Review method: Reviewed against standard ellipse formulas including Ramanujan's approximation and eccentricity definition.

Calculator tool

How this calculator works

Use the explanation to understand the formula, assumptions, and practical limits behind the calculator result.

What an Ellipse Is

An ellipse is a closed oval curve. Every point on the ellipse has the same sum of distances to two fixed points called the foci. A circle is a special ellipse where both foci are at the same point (the center) and the two axes are equal.

Inputs Explained

  • Semi-major axis (aa): half the length of the longest diameter (the widest span).
  • Semi-minor axis (bb): half the length of the shortest diameter. Always b≤ab \leq a.

Formulas

MeasurementFormula
AreaA=πabA = \pi a b
Perimeter (Ramanujan)P≈π(a+b)(1+3h10+4−3h)P \approx \pi(a+b)\left(1 + \dfrac{3h}{10 + \sqrt{4 - 3h}}\right)
Eccentricitye=1−b2/a2e = \sqrt{1 - b^2/a^2}

where h=(a−b)2(a+b)2h = \dfrac{(a-b)^2}{(a+b)^2}

Worked Example

For an ellipse with a=8a = 8 and b=5b = 5:

A=π×8×5≈125.66 sq unitsA = \pi \times 8 \times 5 \approx 125.66\text{ sq units}
e=1−25/64≈0.781e = \sqrt{1 - 25/64} \approx 0.781

Eccentricity ranges from 0 (circle) to just under 1 (very elongated). At e≈0.78e \approx 0.78, this ellipse is moderately elongated.

Eccentricity Explained

Eccentricity (ee)Shape
0Perfect circle
0–0.5Gently oval
0.5–0.9Noticeably elongated
Near 1Very thin, almost like a line

Perimeter Note

Unlike a circle's circumference (C=2πrC = 2\pi r), there is no simple exact formula for an ellipse's perimeter. Ramanujan's approximation (used here) is accurate to within 0.0003% for most ellipses.

Frequently asked questions

What is the difference between semi-major and semi-minor axes?

The semi-major axis (aa) is half the longest diameter — measured from center to the farthest point. The semi-minor axis (bb) is half the shortest diameter. If both are equal, the ellipse is a circle. Enter a≥ba \geq b for correct eccentricity calculation.

Why is there no exact formula for the ellipse perimeter?

The perimeter of an ellipse requires an infinite series to express exactly — it cannot be reduced to a simple combination of aa, bb, and π\pi. Several approximations exist; this calculator uses Ramanujan's second approximation, which is extremely accurate (error below 0.0003%) for ellipses ranging from nearly circular to moderately elongated.

How does eccentricity relate to the shape of an ellipse?

Eccentricity (ee) measures how elongated the ellipse is. At e=0e = 0 the ellipse is a perfect circle. As ee increases toward 1, the ellipse becomes more stretched. Earth's orbit around the sun has e≈0.017e \approx 0.017 (nearly circular). A very elongated comet orbit may have e≈0.99e \approx 0.99.

What real-world shapes are ellipses?

Planetary orbits, the shadow cast by a tilted circle, oval running tracks, elliptical mirrors, the cross-section of a cylinder cut at an angle, and many lenses are ellipses or approximations of them. In architecture, elliptical arches distribute load differently from circular arches.