Calculator Features
📐Three practical solve modes
🧮Step-by-step working
🎯Ramanujan perimeter, verified by numerical integration
🚫Detects impossible & invalid inputs
🔄Unit conversion (linear & area)
🖨Printable / PDF results
📊CSV export
📋Copy results & formula
🎚Decimal precision control
📱Mobile friendly
🧭 Jump to a section
What Is an Ellipse?
An ellipse is the set of all points where the sum of the distances to two fixed points (the foci) is constant. Visually, it's a "stretched circle" — flattened along one direction. Every ellipse has a semi-major axis (a, the longer half-width) and a semi-minor axis (b, the shorter half-width), which together determine everything else about its shape.
💡 Did You Know? Unlike every polygon-based shape on this site, an ellipse's perimeter has no exact formula using ordinary algebra — it genuinely requires calculus (an elliptic integral). This isn't a limitation of this calculator; it's a real mathematical fact discovered by Ramanujan and others, who spent real effort finding extremely accurate approximations instead.
Properties of an Ellipse
- Two axes of symmetry: the major axis (length 2a) and minor axis (length 2b), meeting at the centre at right angles.
- Two foci, located on the major axis at distance c = √(a²−b²) from the centre.
- The sum of distances from any point on the ellipse to both foci is always 2a (constant).
- Eccentricity (e) measures how "stretched" the ellipse is, from 0 (a circle) approaching 1 (very elongated).
- A circle is the special case where a = b, making both foci coincide at the centre.
Ellipse vs Circle
| Property | Ellipse | Circle |
|---|---|---|
| Axes | Two different lengths (a ≠ b) | One radius (a = b) |
| Foci | Two distinct points | Both coincide at the centre |
| Eccentricity | 0 < e < 1 | e = 0 |
| Perimeter formula | No exact elementary formula | Exact: 2πr |
| Relationship | A circle is an ellipse with equal semi-axes | |
See the dedicated circle calculator for that specific case.
Area Formula
A = π × a × b
Simple and exact — a direct generalisation of a circle's A = πr², since a circle is just an ellipse with a = b = r. An ellipse with semi-axes 5 and 3 has an area of π×5×3 ≈ 47.124 square units.
Bounding Box and Area Efficiency
The smallest rectangle that fully contains an ellipse — its bounding box — always has width 2a and height 2b, giving a bounding rectangle area of 2a×2b = 4ab. Comparing that to the ellipse's own area of πab gives a strikingly simple result:
Area efficiency = πab / 4ab = π/4 ≈ 78.54%
An ellipse always fills exactly 78.54% of its bounding rectangle — regardless of how elongated it is. A near-circular ellipse and a very flattened one both occupy precisely that same fraction, since the a and b terms cancel out of the ratio entirely. This is genuinely useful for designers and fabricators estimating material waste when cutting an elliptical piece from rectangular stock.
This calculator also classifies every ellipse by how elongated it is, purely from its eccentricity: 🟢 Circle (e = 0), 🔵 Nearly Circular (e < 0.3), 🟠 Moderately Elongated (0.3 ≤ e < 0.7), or 🔴 Highly Elongated (e ≥ 0.7) — a quick visual read to complement the exact numbers.
Perimeter and Elliptic Integrals
The exact perimeter of an ellipse is P = 4a·E(e), where E(e) is the complete elliptic integral of the second kind — a function with no expression in ordinary algebra, only as an infinite series or an integral that must be evaluated numerically. This calculator uses two independent methods and shows both:
- Ramanujan's approximation: P ≈ π(a+b)(1 + 3h/(10+√(4−3h))), where h = ((a−b)/(a+b))². Published by Srinivasa Ramanujan in 1914, its error is typically smaller than 1 part in a million for any eccentricity.
- Numerical integration: this calculator also evaluates the elliptic integral directly using Simpson's rule, as an independent check on the Ramanujan figure.
For almost every practical purpose these two values agree to more decimal places than any real-world measurement could ever be taken to — this calculator shows both so you can see that agreement for yourself, rather than asking you to trust a single unverified formula.
Eccentricity
e = √(1 − b²/a²)
Eccentricity is a number between 0 and 1 describing how elongated an ellipse is. e = 0 is a perfect circle; as e approaches 1, the ellipse becomes an increasingly flattened sliver. Planetary orbits are ellipses with the Sun at one focus — Earth's orbit has an eccentricity of only about 0.017 (very nearly circular), while a typical comet's orbit can exceed 0.9.
Foci and Directrices
The two foci sit on the major axis at (±c, 0), where c = √(a²−b²) is the linear eccentricity. They have a beautiful reflective property: a ray of light, or a sound wave, leaving one focus and bouncing off the ellipse always arrives at the other focus — the principle behind "whispering gallery" rooms and elliptical reflector telescopes. Each ellipse also has two directrices — lines at x = ±a/e — used in the classical focus-directrix definition of a conic section.
Coordinate Geometry
An ellipse centred at the origin, with its major axis along the x-axis, satisfies the standard equation x²/a² + y²/b² = 1. Every point on it can also be written in parametric form as (a·cos θ, b·sin θ) for θ from 0 to 2π — this calculator uses that parametrisation to draw the scale diagram. The four vertices sit at (±a, 0) and (0, ±b).
Worked Examples
Area from semi-axes — a=5, b=3
A = π×5×3 ≈ 47.124 square units
Perimeter (Ramanujan) — a=5, b=3
h = ((5−3)/(5+3))² = 0.0625
P ≈ π×8×(1 + 3×0.0625/(10+√(4−0.1875))) ≈ 25.527 units
Eccentricity and foci — a=5, b=3
e = √(1−9/25) = √0.36 = 0.8; c = √(25−9) = 4, so foci at (±4, 0)
Missing axis from area — Area=50, a=10
b = Area/(πa) = 50/(π×10) ≈ 1.592 units
Missing axis from eccentricity — e=0.6, a=10
b = a×√(1−e²) = 10×√(1−0.36) = 8 units
Astronomy — a satellite orbit
A low-Earth orbit with semi-major axis 7000 km and eccentricity 0.168 has semi-minor axis ≈ 6899.5 km — nearly circular but not quite
Whispering gallery — architectural acoustics
An elliptical room with a=10 m, b=6 m: e=0.8, foci at ±8 m — sound from one focus point reflects precisely to the other
Impossible-input example
Eccentricity entered as 1.2: eccentricity must be strictly less than 1, since 1 or more describes a parabola or hyperbola, not a closed ellipse
The calculator correctly rejects this rather than returning a meaningless shape
Real-World Applications
| Field | Use |
|---|---|
| Astronomy | Planetary and satellite orbits (Kepler's first law): all orbits are ellipses with the larger body at one focus |
| Architecture | Elliptical windows, domes, and whispering-gallery rooms using the focus-reflection property |
| Engineering | Elliptical gears, cams, and pressure vessel heads |
| Sports & recreation | Athletics tracks and elliptical exercise machines |
| Optics | Elliptical mirrors and reflectors that focus light or sound from one focal point to another |
| Landscaping | Elliptical garden beds, pools, and lawn areas |
Real-World Examples
🛰
Satellite orbits
Kepler's elliptical orbits
Kepler's elliptical orbits
🎤
Whispering galleries
Focus-to-focus acoustics
Focus-to-focus acoustics
🏃
Running tracks
Athletics field design
Athletics field design
🪟
Architecture
Elliptical windows & domes
Elliptical windows & domes
⚙️
Engineering
Elliptical gears & cams
Elliptical gears & cams
🏊
Pools & landscaping
Elliptical pools & gardens
Elliptical pools & gardens
Glossary
- Semi-major axis (a)
- Half the length of the ellipse's longer axis.
- Semi-minor axis (b)
- Half the length of the ellipse's shorter axis.
- Focus (plural: foci)
- One of two special points inside the ellipse; distances to both foci from any point on the ellipse always sum to 2a.
- Eccentricity (e)
- A number from 0 (circle) to just under 1 (very elongated) describing the ellipse's shape.
- Linear eccentricity (c)
- The distance from the centre to each focus, c = √(a²−b²).
- Directrix
- One of two lines (x = ±a/e) used in the classical focus-directrix definition of an ellipse.
- Semi-latus rectum
- Half the length of the chord through a focus, perpendicular to the major axis; equals b²/a.
- Elliptic integral
- A type of integral with no elementary closed-form antiderivative, required to compute an ellipse's exact perimeter.
Common Mistakes
| Mistake | Fix |
|---|---|
| Using P = π(a+b) for perimeter | This crude estimate can be off by several percent for elongated ellipses — use the Ramanujan approximation instead |
| Confusing the full axes (2a, 2b) with the semi-axes (a, b) | All the formulas on this page use the semi-axes; double them for the full major/minor axis lengths |
| Assuming an ellipse's perimeter has an exact formula | It doesn't — only very accurate approximations exist, and this calculator is transparent about that |
| Mixing up eccentricity with linear eccentricity | e is a dimensionless ratio (0 to 1); c is an actual length, in the same units as a and b |
| Forgetting eccentricity must be less than 1 | e ≥ 1 describes a parabola or hyperbola, not a closed ellipse |
Formula Cheat Sheet
Quick Reference
Area:A = πabPerimeter (Ramanujan):
P ≈ π(a+b)(1+3h/(10+√(4−3h))), h=((a−b)/(a+b))²Eccentricity:
e = √(1−b²/a²) | Linear eccentricity: c = √(a²−b²)Foci:
(±c, 0) | Directrices: x = ±a/e | Semi-latus rectum: b²/aStandard equation:
x²/a² + y²/b² = 1 | Parametric form: (a·cos θ, b·sin θ)
Practice Questions
Beginner (with answers)
- Find the area of an ellipse with semi-axes 6 and 4.
- Find the area of an ellipse with semi-axes 10 and 8.
- An ellipse has a=5, b=5. What special shape is it?
- Find the eccentricity of an ellipse with a=10, b=8.
- Find the linear eccentricity (c) of an ellipse with a=13, b=5.
Show answers
1) π×6×4≈75.398 2) π×10×8≈251.327 3) A circle (a=b) 4) e=√(1−64/100)=0.6 5) c=√(169−25)=12
Advanced (with answers)
- Find the perimeter (Ramanujan) of an ellipse with a=6, b=4.
- An ellipse has area 50 and semi-major axis a=10. Find b.
- An ellipse has eccentricity 0.6 and semi-major axis a=10. Find b.
- Find the foci of an ellipse with a=13, b=5.
- Find the semi-latus rectum of an ellipse with a=6, b=4.
Show answers
1) h=(2/10)²=0.04, P≈π×10×(1+0.12/(10+√3.88))≈31.731 2) b=50/(π×10)≈1.592 3) b=10×√(1−0.36)=8 4) c=√(169−25)=12, foci at (±12,0) 5) b²/a=16/6≈2.667
🔑 Key Takeaways
- An ellipse is defined by two semi-axes; area A = πab is simple and exact.
- Perimeter has no elementary closed form — this calculator uses Ramanujan's approximation, verified against numerical integration.
- Eccentricity (0 to 1) measures elongation; e=0 is a circle.
- Foci sit at (±c, 0) where c=√(a²−b²); the sum of focal distances from any point is always 2a.
- A circle is the special case of an ellipse where the two semi-axes are equal.
Frequently Asked Questions
A shape defined as the set of all points where the sum of distances to two fixed points (the foci) is constant — visually, a stretched circle with a longer axis and a shorter axis.
A = π×a×b, using the semi-major axis (a) and semi-minor axis (b). An ellipse with semi-axes 5 and 3 has an area of about 47.124 square units.
There's no exact elementary formula — this calculator uses Ramanujan's highly accurate approximation, P ≈ π(a+b)(1+3h/(10+√(4−3h))), verified against a direct numerical integration of the exact elliptic integral.
The exact perimeter requires an elliptic integral — a type of integral proven to have no expression in elementary functions (no combination of roots, powers, logs or trig functions). It can only be computed via series expansion or numerical integration.
Extremely — its error is typically smaller than one part in a million for any eccentricity. This calculator shows it alongside an independent numerical integration so you can see the two agree.
a is the semi-major axis (half the longer axis) and b is the semi-minor axis (half the shorter axis). If you enter them the wrong way round, this calculator relabels them automatically rather than silently using the wrong one as "major."
A number from 0 to just under 1 describing how elongated an ellipse is. e = √(1−b²/a²); e=0 is a perfect circle, and values approaching 1 describe increasingly flattened ellipses.
Two special points on the major axis, at (±c, 0) where c=√(a²−b²). The sum of the distances from any point on the ellipse to both foci is always exactly 2a.
One of two lines, at x=±a/e, used in the classical focus-directrix definition of an ellipse: the ratio of a point's distance to a focus versus its distance to the corresponding directrix always equals the eccentricity.
A circle is a special ellipse where both semi-axes are equal (a=b), making its eccentricity exactly 0 and its two foci coincide at the centre. A circle's perimeter also has a simple exact formula (2πr), unlike a general ellipse.
Yes — mathematically, a circle is the special case of an ellipse where a=b.
x²/a² + y²/b² = 1, for an ellipse centred at the origin with its major axis along the x-axis.
(a·cos θ, b·sin θ), for θ ranging from 0 to 2π — this traces out every point on the ellipse as θ sweeps around a full circle.
Rearranging A=πab: the missing axis equals Area divided by (π times the known axis).
If a is known: b = a×√(1−e²). If b is known: a = b/√(1−e²).
Not directly as a solve mode on this calculator — because the perimeter formula only has an approximate (not exact) inverse, we've kept it as a value you can check your entered semi-axes against, rather than solve backward from, to avoid presenting an approximation as if it were an exact input.
Half the length of the chord passing through a focus, perpendicular to the major axis. It equals b²/a, and is useful in orbital mechanics and optics.
Kepler's first law states that planetary orbits are ellipses with the Sun at one focus. Earth's orbit is nearly circular (eccentricity ≈ 0.017), while comets can have eccentricities above 0.9.
Elliptical "whispering gallery" rooms use the focus-reflection property — sound leaving one focus reflects off the walls and arrives clearly at the other focus, even across a large room.
Elliptical gears and cams convert rotational motion into variable-speed motion, and elliptical heads are used on pressure vessels for their strength-to-weight properties.
Any linear unit for the semi-axes, linear eccentricity and directrix distance (mm, cm, m, ft); area uses the squared version of the same unit (m², ft²); eccentricity has no unit.
Yes — choose Semi-Axes, Area + Axis, or Eccentricity + Axis above, enter the known values, and the calculator solves everything else instantly with the working shown.
Area, eccentricity, foci and directrices are calculated exactly (to floating-point precision). Perimeter uses the Ramanujan approximation, cross-checked against numerical integration, and is accurate to many more decimal places than any physical measurement could achieve.
Yes — enter the known values for whichever mode matches, and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
Using the crude P=π(a+b) perimeter estimate instead of a proper approximation; confusing the full axis lengths (2a, 2b) with the semi-axes (a, b); and assuming an exact perimeter formula exists when it doesn't.
The calculator uses the required values for the selected mode to solve the shape, and cross-checks any extra values you entered (perimeter or area) against the result, flagging a warning if they don't match.
Yes — use Copy Results to copy everything to the clipboard, Copy Formula for just the formulas used, Export CSV for a spreadsheet-ready file, or Print / Save as PDF for a printable worksheet.
An ellipse only has two independent measurements (its two semi-axes), so any two independent pieces of information determine it fully. Each mode uses a natural, unambiguous combination — direct axes, area plus one axis, or eccentricity plus one axis.
The calculator automatically relabels them so a is always the longer (semi-major) axis, and tells you this happened in the step-by-step working, rather than silently computing the wrong eccentricity or foci.
Yes — an eccentricity of exactly 0 means a=b, which is a circle.
No — for a closed ellipse, eccentricity must be strictly between 0 and 1. An eccentricity of exactly 1 describes a parabola, and values above 1 describe a hyperbola — neither is a closed ellipse, and this calculator rejects those values.
No — this calculator covers the 2D ellipse only. An ellipsoid (the 3D equivalent) has three semi-axes and different volume and surface-area formulas.
An ellipse doesn't have diagonals in the polygon sense — its two axes (major and minor) play that structural role instead, always perpendicular and meeting at the centre.
This calculator uses Simpson's rule with 2,000 intervals, which typically agrees with the Ramanujan approximation to around 10 significant figures — far beyond what any practical application would need.
To be transparent: one is the fast Ramanujan approximation, the other is a genuine numerical integration of the exact elliptic integral. Showing both, rather than just one unverified number, lets you see directly how closely they agree.
There's no fixed minimum size — any positive semi-axes describe a valid ellipse. The only real constraint is on eccentricity, which must stay strictly between 0 and 1.
Always exactly π/4 ≈ 78.54%, no matter how elongated the ellipse is. The bounding box area is 2a×2b = 4ab, and the ellipse area is πab, so the ratio πab/4ab = π/4 is constant — the a and b terms cancel out completely.
They're a quick visual read of eccentricity: Nearly Circular is e below 0.3, Moderately Elongated is 0.3 up to 0.7, and Highly Elongated is 0.7 and above, with e=0 being a perfect circle. They're a plain-English complement to the exact eccentricity value, not a replacement for it.
Calculation Assumptions
ℹ️ What this calculator assumes
- Results are calculated using Euclidean (flat-plane) geometry.
- Measurements are assumed to lie in a flat plane, not on a curved or sloped surface.
- Rounding to the selected decimal precision may cause very small differences between displayed values and hand calculations.
- All input values are assumed to use the same unit — mixing units (e.g. a semi-major axis in metres and a semi-minor axis in centimetres) will produce an incorrect result.
- Perimeter is calculated using the Ramanujan II approximation, cross-checked against a genuine numerical integration of the exact elliptic integral — no exact elementary formula exists for an ellipse's perimeter.
About This Calculator
Printable Formula Sheet
A one-page reference with every ellipse formula on this page.
Ellipse Formula Sheet
MegaCalcOnline.com · Area, perimeter, eccentricity and foci formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Area | A = πab |
| Perimeter (Ramanujan) | P ≈ π(a+b)(1+3h/(10+√(4−3h))), h=((a−b)/(a+b))² |
| Eccentricity | e = √(1−b²/a²) |
| Linear eccentricity | c = √(a²−b²) |
| Foci | (±c, 0) |
| Directrices | x = ±a/e |
| Semi-latus rectum | b²/a |
| Standard equation | x²/a² + y²/b² = 1 |
References
- Khan Academy — Conic sections and ellipses
- OpenStax Mathematics — analytic geometry
- ACARA — Australian Curriculum: Mathematics
- NIST — SI units for length and area
- Encyclopaedia Britannica — Ellipse
- Wolfram MathWorld — Ellipse
Every formula and worked example on this page was independently verified — see “About This Calculator” above for the full review notes.