Ellipse Calculator — area, circumference, eccentricity and foci

The ellipse calculator above takes the two axes of an ellipse and returns its area, circumference, eccentricity and focal geometry. It accepts either semi-axes or full axes, because drawings and physical measurements give you different things — a CAD file usually specifies semi-axes, while a tape across a real oval gives you the full width and height.

One point matters more than any other on this page, and Arb Digital would rather say it up front than bury it: the area of an ellipse has a clean exact formula, but the circumference does not. There is no exact expression for an ellipse's perimeter using ordinary algebraic functions. The figure this tool reports is Ramanujan's approximation, and the section below explains exactly how close it is.

What This Ellipse Calculator Does

Enter a semi-major axis of 5 cm and a semi-minor axis of 3 cm and the tool returns an area of about 47.12 cm², a circumference of about 25.53 cm, an eccentricity of 0.8, a centre-to-focus distance of 4 cm and a semi-latus rectum of 1.8 cm. Switch to full-axis mode and the same ellipse is described as 10 cm by 6 cm — the numbers change, the shape does not.

If you enter two equal values, the ellipse is a circle. The calculator still works and reports an eccentricity of zero, but for round shapes our circle calculator gives more relevant outputs, including an exact circumference with no approximation involved.

How to Use It

1. Decide whether you have semi-axes or full axes. If you measured across the whole oval, you have full axes; halve them or switch the mode.
2. Enter the longer value first. The tool sorts them internally, but keeping major and minor in the right boxes makes the working easier to read.
3. Select a unit so the area is labelled with the correct squared unit.
4. Read the eccentricity to judge the shape: 0 is a circle, values approaching 1 are increasingly flattened.
5. Check the working line, which shows the area formula and the Ramanujan circumference calculation using your own numbers.

The Formula: Area Is Exact, Circumference Is Not

The area of an ellipse is A = πab, where a and b are the semi-major and semi-minor axes. With a = 5 and b = 3, A = π × 15 ≈ 47.1239 cm². The formula is a direct generalisation of the circle: set a = b = r and it collapses to πr². Geometrically, an ellipse is a circle stretched by a factor a/b along one direction, and stretching multiplies area by the same factor, which is why the result is this simple.

The circumference is another matter entirely. Working it out exactly requires a complete elliptic integral of the second kind, which cannot be written in terms of elementary functions — no combination of powers, roots, logs and trig functions produces it. This is a proven limitation of the problem, not a gap in anyone's algebra. MathWorld's ellipse entry sets out the integral form and the approximations built to stand in for it.

Ramanujan's Approximation and How Accurate It Really Is

This calculator uses Ramanujan's second approximation, published in 1914:

C ≈ π(a + b)[1 + 3h / (10 + √(4 − 3h))], where h = ((a − b)/(a + b))²

Work it through for a = 5, b = 3. First h = (2/8)² = 0.0625. Then 4 − 3h = 3.8125, whose square root is 1.952562. The bracket is 1 + (0.1875 / 11.952562) = 1.015687. Multiply by π(a + b) = π × 8 = 25.13274, giving C ≈ 25.5267 cm. The true value, computed from the elliptic integral, is 25.52699 — the approximation is accurate here to about one part in a million.

The error is not constant. It depends only on how elongated the ellipse is, and it grows as the shape flattens. For anything close to circular it is negligible at any practical precision. The worst case is the degenerate limit where b approaches zero and the ellipse collapses to a line segment of length 2a, which has to be traced out and back for a total of 4a. There Ramanujan's formula gives π × a × (14/11) ≈ 3.99845a against a true value of 4a — an error of about 0.039%. So across the entire range of possible ellipses, from perfect circle to flattened line, this approximation is never worse than four hundredths of one percent, and is usually far better than that. That is why we present it as an approximation rather than an exact answer, and also why it is more than good enough for any physical job.

Eccentricity: The Number That Describes the Shape

Eccentricity measures how far an ellipse departs from being a circle: e = √(1 − b²/a²). For a = 5 and b = 3, e = √(1 − 9/25) = √0.64 = 0.8. A value of 0 is a perfect circle; values approaching 1 describe increasingly stretched ovals; e = 1 is the degenerate line segment.

Eccentricity is scale-free, which makes it the right way to compare shapes of different sizes. Two ellipses with the same eccentricity are the same shape at different magnifications, exactly as two rectangles with the same aspect ratio are. It is also the standard descriptor in orbital mechanics. NASA's planetary fact sheets list Earth's orbital eccentricity as 0.0167, which is so close to zero that the orbit is visually indistinguishable from a circle — a useful corrective to the exaggerated ovals in most textbook diagrams.

Foci, and the String-and-Pins Definition

An ellipse has two foci, sitting on the major axis at a distance c = √(a² − b²) from the centre. For our example, c = √(25 − 9) = 4 cm, so the foci are 8 cm apart. The defining property is that for every point on the curve, the sum of the distances to the two foci is constant and equal to 2a.

That is not just theory — it is the practical construction method. Push two pins into a board 2c apart, tie a loop of string of length 2a + 2c around them, and trace with a pencil held taut. The result is a true ellipse with semi-major axis a. Gardeners use this to lay out oval beds, and it is still the fastest way to mark an accurate large ellipse without a template. This calculator gives you the two numbers the method needs: the pin separation and the semi-major axis.

The final grid figure, the semi-latus rectum, is b²/a — for our example, 9/5 = 1.8 cm. It is the half-width of the ellipse measured vertically at a focus, and it is the parameter that appears in the orbital form of the ellipse equation.

Why Averaging the Axes Does Not Work

The instinctive shortcut for an ellipse circumference is to average the two semi-axes and treat the shape as a circle of that radius: C ≈ 2π × (a + b)/2 = π(a + b). It is tempting because it is the leading term of Ramanujan's formula, and for a near-circular ellipse it is genuinely close. The problem is that it is always an underestimate, and the shortfall grows fast.

For our 5 by 3 example, π(a + b) = 25.133 against a true 25.527 — about 1.5% low. Push the ellipse to 5 by 1 and the mean-radius estimate gives 18.85 while the true perimeter is about 21.01, an error of over 10%. In the flat limit it gives πa ≈ 3.1416a against the true 4a, a 21% shortfall. So the shortcut is fine for a shape you can see is almost round and useless for anything noticeably elongated, which is precisely when you were most likely to reach for it.

A second common shortcut, using the geometric mean √(ab) as the radius, is worse still: it underestimates by more than the arithmetic mean does at every eccentricity. Ramanujan's expression exists precisely because these easy substitutions fail in the region where accuracy matters, and it costs nothing extra to compute.

Where Ellipses Turn Up

Cut a cylinder at an angle and the cross-section is an ellipse — which is why every angled pipe joint, duct transition and mitred tube produces an elliptical opening rather than a circular one. If you are marking out sheet metal to wrap such a joint, the ellipse dimensions come from the pipe diameter and the cut angle: the minor axis stays equal to the diameter while the major axis stretches by 1/cos of the angle.

The same shape appears in whispering galleries, where sound from one focus reflects to the other; in elliptical machine trainers; in oval running tracks, though those are usually two straights joined by semicircles rather than true ellipses; and in every planetary orbit, following Kepler's first law. A shadow cast by a circular object on an angled surface is also an ellipse, which is the everyday version of the cylinder cut.

For the round version of any of these calculations use the circle calculator, and for a straight-sided outline the polygon calculator or rectangle area calculator will be closer to what you need.

Common Mistakes to Avoid

• Mixing semi-axes with full axes — using a full axis where a semi-axis belongs quadruples the area. Use the mode selector rather than halving in your head.
• Treating the reported circumference as exact — it is an approximation, accurate but not exact, because no exact elementary formula exists.
• Averaging the two axes and using the circle formula — π × mean radius understates the true perimeter, and the error grows quickly with eccentricity.
• Confusing the semi-minor axis with the focal distance — b and c are different quantities related by a² = b² + c².
• Assuming an oval is an ellipse — stadium shapes, egg shapes and rounded rectangles are ovals but not ellipses, and none of these formulas apply to them.

Related Free Tools From Arb Digital

Handle the circular case with the circle calculator, measure straight-edged boundaries with the perimeter calculator, work out any regular shape with the polygon calculator, convert a finished area with the area converter, and convert linear dimensions with the length converter. Everything we publish is listed in the free online tools hub.

Frequently Asked Questions