Ellipse Calculator | Area, Perimeter & Eccentricity
Calculate the area, perimeter (Ramanujan II), eccentricity, focal distance, semi-latus rectum, directrix, flattening, and standard equation of any ellipse from its semi-axes.
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What Is the Ellipse Calculator | Area, Perimeter & Eccentricity?
An ellipse is the set of all points P in a plane such that the sum of distances from P to two fixed points, the foci, is constant and equal to 2a. This definition gives the ellipse its characteristic oval shape and explains why the semi-major axis a is always half of that constant sum. When the two foci coincide (c = 0), the ellipse degenerates into a circle of radius a = b.
The area formula A = πab is exact and follows from the fact that an ellipse is a circle of radius a scaled by factor b/a along the minor axis. The perimeter, however, has no exact closed form in elementary functions, it involves a complete elliptic integral of the second kind. Ramanujan's second approximation (used by this calculator) has an error below 0.001% across the full practical range of eccentricities, making it the standard choice for engineering and scientific work.
Eccentricity e = c/a is the single number that characterises the shape of an ellipse regardless of its size. It ranges from 0 (perfect circle) to just below 1 (nearly degenerate flat ellipse). Kepler's first law states that all planetary orbits are ellipses with the Sun at one focus, Earth's orbital eccentricity is 0.0167 (nearly circular), while Halley's Comet has e ≈ 0.967 (extremely elongated).
The semi-latus rectum ℓ = b²/a is the half-chord drawn through a focus parallel to the minor axis. It appears naturally in the polar equation of the ellipse (r = ℓ/(1 − e cos θ)) and is important in orbital mechanics for relating orbital speed to position. The directrix, at distance a/e from the centre, is the line such that the ratio of a point's distance to a focus and to the nearest directrix equals e.
Formula
| Symbol | Name | Description |
|---|---|---|
| a | Semi-major axis | Half the longest diameter; always ≥ b |
| b | Semi-minor axis | Half the shortest diameter; always ≤ a |
| c | Focal distance | √(a²−b²); distance from centre to each focus |
| e | Eccentricity | c/a; 0 for circle, approaches 1 for a very flat ellipse |
| h | Ramanujan h | (a−b)²/(a+b)²; used in the perimeter approximation |
| ℓ | Semi-latus rectum | b²/a; half-chord through a focus parallel to minor axis |
| f | Flattening | (a−b)/a; 0 for circle, approaches 1 for flat ellipse |
| x/a | Directrix distance | a/e = a²/c; distance from centre to each directrix line |
How to Use
- 1Choose input mode: "Semi-axes (a, b)" for the standard half-lengths, or "Full axes (2a, 2b)" if you measured the total major and minor diameters directly.
- 2Load a preset: Click any preset, classic geometry (a=5, b=3), circle, Earth or Mars orbit, Halley's Comet, or an oval track, to populate values and calculate immediately.
- 3Enter your values: Type the semi-major axis a (larger value) and semi-minor axis b (smaller value). Decimals are fully supported. The calculator auto-corrects if you enter b > a.
- 4Press Calculate or Enter: Click "Calculate Ellipse" or press Enter from any input. An interactive SVG diagram appears alongside all computed properties.
- 5Read the diagram: The SVG shows the ellipse with labelled semi-axes a and b, the two foci (F₁, F₂), and the focal distance c. The legend indicates a circle when a = b.
- 6Check all properties: The primary card shows area, perimeter, eccentricity, and focal distance. The grid below adds semi-latus rectum, flattening, directrix, aspect ratio, and full axis lengths.
- 7Expand calculation steps: Click the steps panel to see every formula substituted with your specific numbers, useful for checking homework or exam working.
- 8Copy or reset: Click "Copy results" to copy all values to the clipboard. Press Reset or Esc to clear. Your last inputs are saved to browser storage and restored on your next visit.
Example Calculation
Example 1: Classic geometry, a = 5, b = 3
Example 2: Earth's orbit (semi-axes in AU)
Earth's orbital semi-major axis is 1.000 AU; semi-minor axis 0.9998 AU.
Example 3: Highly elongated, Halley's Comet (AU)
Halley's Comet: a ≈ 17.83 AU, b ≈ 4.57 AU (e ≈ 0.967).
Understanding Ellipse | Area, Perimeter & Eccentricity
What Is an Ellipse?
An ellipse is a closed curve that belongs to the family of conic sections, shapes formed by slicing a cone with a flat plane. Tilt the plane at a shallower angle than the cone's surface and the cross-section is an ellipse. Steeper angles give a hyperbola; a parallel-to-side angle gives a parabola; a horizontal slice gives a circle (the special case where e = 0).
The formal definition is purely distance-based: an ellipse is all points P such that PF₁ + PF₂ = 2a, where F₁ and F₂ are the two foci. This definition requires no coordinates, it explains why planetary orbits are ellipses (gravity provides the required constraint), why elliptical mirrors focus all rays from one focus to the other, and why satellite dishes and lithotripters exploit the same focusing geometry.
Why Is There No Exact Perimeter Formula?
The arc length of an ellipse is given by the integral L = 4a ∫₀^(π/2) √(1 − e² sin²θ) dθ, which is a complete elliptic integral of the second kind. This class of integrals has been proven to have no closed form expressible in terms of elementary functions (polynomials, exponentials, trigonometric functions). This is a deep result from the theory of algebraic functions, not just a limitation of current mathematical techniques.
Ramanujan, working in the early 20th century, found two remarkable rational approximations. His second formula, used by this calculator, achieves accuracy better than 1 part in 100,000 for all practical eccentricities. For a circle (e = 0) and for e → 1 both approximations become exact; the maximum error of Ramanujan II occurs around e ≈ 0.56 and is about 0.00007%.
Eccentricity Explained
- ›e = 0: perfect circle. The two foci coincide at the centre. a = b.
- ›0 < e < 0.5: nearly circular ellipse. Planetary orbits mostly fall here (Earth e = 0.017, Venus e = 0.007).
- ›0.5 ≤ e < 0.9: noticeably elongated. Pluto's orbit (e = 0.249) and Mercury (e = 0.206) fall in this range.
- ›0.9 ≤ e < 1: highly elongated. Comets often have eccentricities in this range (Halley: e = 0.967).
- ›e = 1: degenerate case, the ellipse opens into a parabola. No longer a closed curve.
The Semi-Latus Rectum in Orbital Mechanics
The semi-latus rectum ℓ = b²/a has a beautiful geometric meaning: it is the distance from a focus to the ellipse, measured perpendicular to the major axis. In orbital mechanics, it enters the polar equation of the orbit: r = ℓ / (1 − e cos θ), where θ is the angle from periapsis (closest point) and r is the distance from the focus (the body being orbited).
This form makes Kepler's first law immediately usable: given the semi-latus rectum and eccentricity of any orbit, you can compute the distance from the central body at any orbital angle. At periapsis (θ = 0): r = ℓ/(1+e) = a(1−e). At apoapsis (θ = π): r = ℓ/(1−e) = a(1+e).
Real-World Applications
| Field | Application | Key Property Used |
|---|---|---|
| Astronomy | Planetary and cometary orbits (Kepler's 1st Law) | Semi-major axis a, eccentricity e |
| Optics | Ellipsoidal reflectors, James Webb telescope mirrors | Focus-to-focus reflection |
| Medicine | Lithotripsy, focusing ultrasound to break kidney stones | Foci of an ellipsoidal bowl |
| Architecture | Whispering galleries (St Paul's, US Capitol), stadiums | Acoustic focus between foci |
| Engineering | Pressure vessel cross-sections, elliptical gears | Area and stress distribution |
| CAD / Graphics | Bézier ellipses, perspective circles in technical drawing | Exact area and perimeter |
| Geodesy | Earth modelled as an oblate ellipsoid of revolution | Flattening f = (a−b)/a |
The Earth as an Ellipsoid
Earth is not a sphere but an oblate spheroid, an ellipse rotated about its minor axis. The WGS 84 geodetic standard (used by GPS) defines Earth's semi-major axis as a = 6,378.137 km and semi-minor axis as b = 6,356.752 km, giving a flattening of f = 1/298.257 ≈ 0.003353. This small but significant difference affects navigation over long distances and explains why buildings appear to lean in perspective satellite imagery taken from low-Earth orbit. All modern GPS receivers account for the ellipsoidal Earth model when converting coordinates to distances.
Frequently Asked Questions
What is the difference between semi-major and semi-minor axis?
The two axes of an ellipse are its two perpendicular lines of symmetry:
- • Semi-major axis a: half the length of the longest diameter. Runs from the centre to the widest point of the ellipse.
- • Semi-minor axis b: half the length of the shortest diameter. Always perpendicular to a.
- • The full major and minor diameters (sometimes called conjugate diameters) are simply 2a and 2b.
- • When a = b, the two axes are equal and the ellipse is a circle of radius a.
This calculator accepts either form, you can enter semi-axes or full diameters using the toggle at the top.
Why is there no exact formula for ellipse perimeter?
The perimeter of an ellipse requires an elliptic integral, a class of integrals with no closed-form expression in terms of elementary functions. This has been mathematically proven, not merely conjectured.
The exact integral is:
P = 4a ∫₀^(π/2) √(1 − e² sin²θ) dθ
- • Ramanujan I: P ≈ π[3(a+b) − √((3a+b)(a+3b))]. Simple but less accurate.
- • Ramanujan II: P ≈ π(a+b)(1 + 3h/(10+√(4−3h))). Error < 0.001% for nearly all practical ellipses.
This calculator uses Ramanujan II, which is the standard approximation in engineering and scientific software.
What is eccentricity and what values does it take?
Eccentricity e measures how much an ellipse deviates from a circle. It is always between 0 and 1 (exclusive for a proper ellipse):
- • e = 0, perfect circle (a = b, foci coincide at centre)
- • 0 < e < 1, ellipse (all planetary orbits fall here)
- • e = 1, parabola (degenerate open curve, not an ellipse)
- • e > 1, hyperbola (two separate open curves)
Real examples: Earth e ≈ 0.017, Mars e ≈ 0.093, Halley's Comet e ≈ 0.967, Earth's shape (flattening) f ≈ 0.003353.
What are the foci of an ellipse and why do they matter?
The foci (singular: focus) are two special points inside the ellipse, each at distance c = √(a²−b²) from the centre along the major axis.
Their defining property: for every point P on the ellipse, the sum PF₁ + PF₂ = 2a is constant. This has remarkable physical consequences:
- • Optics: a ray of light leaving one focus reflects off the ellipse and passes exactly through the other focus. Ellipsoidal mirrors exploit this for surgical lighting, lithotripsy, and telescope secondaries.
- • Orbital mechanics: planets orbit the Sun, which sits at one focus. The other focus is empty (a mathematical point in space).
- • Acoustics: sound waves from one focus of an elliptical room reconverge at the other, the whispering gallery effect in St Paul's Cathedral and the US Capitol.
How do I find the equation of an ellipse?
The standard equation of an ellipse centred at the origin with horizontal major axis is:
x²/a² + y²/b² = 1
- • Horizontal ellipse (a > b): major axis is along the x-axis; vertices at (±a, 0); foci at (±c, 0).
- • Vertical ellipse (b > a): swap a and b in the formula; major axis is along the y-axis.
- • Shifted ellipse: replace x with (x−h) and y with (y−k) for centre at (h, k).
- • The calculator shows the standard equation for your specific a and b values in the results.
What is the semi-latus rectum and when is it used?
The semi-latus rectum ℓ = b²/a is the distance from a focus to the ellipse, measured perpendicular to the major axis. It appears in the polar form of the ellipse equation:
r = ℓ / (1 − e cos θ)
- • This is the standard form in orbital mechanics, where r is the distance from the central body and θ is the angle from periapsis.
- • At periapsis (θ=0): r = a(1−e), closest approach.
- • At apoapsis (θ=π): r = a(1+e), farthest point.
- • For Earth: ℓ ≈ 0.9997 AU; at aphelion r ≈ 1.017 AU, at perihelion r ≈ 0.983 AU.
How is an ellipse different from an oval?
In everyday language, "oval" and "ellipse" are often used interchangeably, but they have different mathematical meanings:
- • Ellipse: a precise mathematical curve defined by the constant-sum-of-distances property. It has exactly two axes of symmetry and is described by the equation x²/a² + y²/b² = 1.
- • Oval: any smooth, closed, convex curve that looks like an egg or a running track. Ovals need not be symmetric, and most are not ellipses.
- • Egg shape: a specific oval with one end more pointed than the other, definitely not an ellipse (which is symmetric).
- • Athletic "oval" running tracks combine two straight sections with two semicircular ends, again, not an ellipse.
An ellipse is the only conic section with a closed form (other than a circle), which is why it arises so naturally in physics and geometry.
Does this calculator save my inputs between visits?
Yes, all inputs are automatically saved to your browser's localStorage as you type:
- • Semi-major axis a, semi-minor axis b, and the chosen input mode are persisted.
- • All data stays in your browser, nothing is sent to any server.
- • When you return to the page, your last inputs are restored automatically.
- • Click Reset or press Esc to clear both the form and the saved data.