What is the eccentricity of an ellipse?

Eccentricity (e) measures how stretched or elongated an ellipse is: e = √(1 − b²/a²), where a is the semi-major axis and b is the semi-minor axis. Eccentricity ranges from 0 (perfect circle, a = b) to just below 1 (extremely elongated, nearly a line). A value of 0.5 means a moderately oval shape. Earth's orbit has eccentricity ≈ 0.0167 (nearly circular).

What is the difference between the major axis and the semi-major axis?

The major axis is the longest diameter of the ellipse, passing through both foci. Its length is 2a. The semi-major axis (a) is half of this - the distance from the centre to the farthest point on the ellipse. Similarly, the minor axis has length 2b, and the semi-minor axis b is its half. The semi-major axis is what you enter in the calculator.

What is the semi-latus rectum of an ellipse?

The semi-latus rectum (l) is the half-length of the chord through one focus perpendicular to the major axis: l = b²/a. It is used extensively in orbital mechanics - for a planet orbiting the Sun, the semi-latus rectum determines the orbit's closest approach (perihelion) and farthest point (aphelion) in terms of the eccentricity.

What is the relationship between a, b, and c in an ellipse?

The three key lengths of an ellipse satisfy the Pythagorean-like identity: a² = b² + c², or equivalently c² = a² − b². Here, c is the focal distance (distance from centre to each focus). This means a is always the hypotenuse of the right triangle formed by b, c, and a. You can rearrange to find any one given the other two: b = √(a² − c²), c = √(a² − b²).

How is the ellipse used in real-world applications?

Ellipses appear throughout science and engineering: planetary orbits (Kepler's first law), satellite and comet trajectories, whispering galleries (the Statuary Hall in the US Capitol), lithotripsy (shock wave therapy using elliptical reflectors), elliptical gears in bicycles, and optical mirrors in telescopes. The reflective property - that a ray from one focus reflects to the other - is the basis for many of these applications.

What does an eccentricity of 0.9 look like?

At e = 0.9, the ellipse is very elongated - more like a rugby ball or a narrow oval. Calculating: c = a × e = 0.9a, and b = a × √(1 − e²) = a × √(1 − 0.81) = 0.436a. So the minor axis is less than half the major axis in length. Comets that barely return from the outer solar system can have eccentricities of 0.99 or higher.

Ellipse Calculator

Q: What is the formula for the perimeter of an ellipse?

There is no simple closed-form formula for the perimeter of an ellipse. The exact value involves an elliptic integral. The best practical approximation is Ramanujan's second formula: P ≈ π × [3(a+b) − √((3a+b)(a+3b))], which is accurate to within 0.0001% for all ellipses. Some simpler but less accurate approximations include P ≈ 2π√((a²+b²)/2).

Q: What are the foci of an ellipse?

An ellipse has two foci (singular: focus) located along the major axis at distance c = √(a² − b²) from the centre. For a horizontally oriented ellipse, the foci are at (±c, 0). A key defining property: for any point P on the ellipse, the sum of distances from P to both foci equals 2a. This property is used in whispering gallery architecture, satellite dish design, and orbital mechanics.

Q: How does an ellipse differ from a circle?

A circle is a special case of an ellipse where the two axes are equal (a = b = r). In a circle, the two foci coincide at the centre, and eccentricity = 0. As a and b diverge, the ellipse becomes more elongated. Computationally, you can treat a circle as an ellipse with a = b - all ellipse formulas degenerate correctly to the circle formulas in that case.

Compute area, perimeter, eccentricity, and foci of any ellipse from its semi-major axis a and semi-minor axis b.

⬭ What is an Ellipse?

An ellipse is a closed, oval-shaped curve in a plane defined as the set of all points where the sum of distances to two fixed points (the foci) is constant and equal to 2a, where a is the semi-major axis. Ellipses are one of the four conic sections - the shapes formed by slicing a cone at various angles - alongside circles (a special ellipse), parabolas, and hyperbolas.

Ellipses govern some of the most important phenomena in nature and engineering. Kepler's first law of planetary motion states that every planet orbits the Sun in an ellipse with the Sun at one focus. The same principle applies to moons, satellites, and comets. Satellite dishes, car headlights, and medical lithotripters exploit the ellipse's reflective property: any ray emitted from one focus reflects off the boundary and converges at the other focus. Whispering galleries - such as the US Capitol's Statuary Hall - are elliptical rooms where a whisper at one focus can be heard clearly at the other, tens of metres away.

The key parameters of an ellipse are: semi-major axis a (half the longest diameter), semi-minor axis b (half the shortest diameter), and focal distance c (distance from centre to each focus), related by a² = b² + c². Eccentricity e = c/a measures how elongated the ellipse is - 0 for a perfect circle and approaching 1 for an extremely narrow oval.

This calculator computes all derived properties - area, perimeter (using Ramanujan's highly accurate approximation), eccentricity, foci coordinates, and semi-latus rectum - from just two inputs: a and b.

📐 Formula

Area = π × a × b

a = semi-major axis (longer radius)

b = semi-minor axis (shorter radius)

Example: a = 6, b = 4 → Area = π × 6 × 4 = 75.40 units²

Perimeter ≈ π × [3(a+b) − √((3a+b)(a+3b))]

Ramanujan's formula - accurate to within 0.0001% for all ellipses

Example: a = 6, b = 4 → P ≈ π × [3(10) − √(22×18)] = π × [30 − 19.90] ≈ 32.66 units

Eccentricity = √(1 − b² ÷ a²) = c ÷ a

c = focal distance = √(a² − b²)

e ranges from 0 (circle) to <1 (elongated ellipse)

Semi-latus rectum = b² ÷ a

Example: a = 6, b = 4 → c = √(36−16) = √20 = 4.47; e = 4.47/6 = 0.745

📖 How to Use This Calculator

Steps

Enter a - the semi-major axis, i.e., half the longest width of your ellipse. This must be the larger of the two axes.

Enter b - the semi-minor axis, i.e., half the shortest width. If you accidentally enter b > a, the calculator swaps them automatically.

Click Calculate - read the area, perimeter, eccentricity, focal distance c, foci positions, and semi-latus rectum in your input units.

💡 Example Calculations

Example 1 — Garden Pond (a = 5 m, b = 3 m)

Find the area and perimeter of an elliptical pond with semi-major axis 5 m and semi-minor axis 3 m.

Area = π × 5 × 3 = 47.12 m².

Perimeter (Ramanujan): h = ((5−3)/(5+3))² = (2/8)² = 0.0625; P ≈ π(5+3)(1 + 3h/(10+√(4−3h))) ≈ 25.53 m.

Eccentricity = √(1 − 9/25) = √(0.64) = 0.800.

Area = 47.12 m² | Perimeter ≈ 25.53 m | e = 0.800

Try this example →

Example 2 — Stadium Track Inner Boundary

A track has a semi-major axis of 85 m and a semi-minor axis of 36 m. What is the enclosed area and perimeter?

Area = π × 85 × 36 = 9,613 m².

Perimeter ≈ 380 m (Ramanujan formula).

Eccentricity = √(1 − 36²/85²) = √(1 − 0.179) = √0.821 = 0.906.

Area ≈ 9,613 m² | Perimeter ≈ 380 m | e = 0.906

Try this example →

Example 3 — Near-Circle (a = 10, b = 9.9)

How do the properties of a near-circular ellipse (a=10, b=9.9) compare to a true circle of radius 10?

Area = π × 10 × 9.9 = 311.02 units² (circle: π × 100 = 314.16 - only 1% smaller).

Perimeter ≈ 62.84 units (circle: 2π × 10 = 62.83 - nearly identical).

Eccentricity = √(1 − 98.01/100) = √0.0199 = 0.141 - very low, almost circular.

Area = 311.02 | Perimeter ≈ 62.84 | e = 0.141

Try this example →

❓ Frequently Asked Questions

What is the formula for the area of an ellipse?+

Area = π × a × b, where a is the semi-major axis (half the longest diameter) and b is the semi-minor axis (half the shortest diameter). This generalises the circle area formula A = πr², which is the special case where a = b = r. For example, an ellipse with a = 6 and b = 4 has area = π × 6 × 4 ≈ 75.40 square units.

What is the formula for the perimeter of an ellipse?+

There is no simple closed-form formula - the exact value requires an elliptic integral. The best practical approximation is Ramanujan's second formula: P ≈ π[3(a+b) − √((3a+b)(a+3b))], accurate to within 0.0001% for all ellipses. A simpler but less accurate estimate is P ≈ 2π√((a²+b²)/2). For nearly circular ellipses, 2π√((a²+b²)/2) is quite good; for very elongated ones, Ramanujan's formula is essential.

What is eccentricity and how is it calculated?+

Eccentricity e = c/a = √(1 − b²/a²), where c is the focal distance. It ranges from 0 (circle) to just below 1 (extremely elongated). An eccentricity of 0.5 produces a moderately oval shape. Earth's orbital eccentricity is 0.0167 (nearly circular); Pluto's is 0.248; Halley's Comet is 0.967 - an extremely elongated orbit reaching far into the outer solar system.

What are the foci of an ellipse?+

The foci (singular: focus) are two special points on the major axis at distance c = √(a² − b²) from the centre. For any point P on the ellipse, the sum of distances from P to both foci equals 2a - this is the defining property. Foci are essential to Kepler's laws of planetary motion, elliptical mirror design, and whispering gallery acoustics.

What is the difference between the major axis and semi-major axis?+

The major axis is the full longest diameter, with length 2a. The semi-major axis a is half of it - the distance from the centre to the farthest point. Similarly, the minor axis has length 2b, and the semi-minor axis b is its half. In this calculator, you enter a and b (the semi-axes), not the full diameters.

How does an ellipse differ from a circle?+

A circle is a special ellipse with a = b = r. Its eccentricity is 0 and its two foci coincide at the centre. All circle formulas are limiting cases of ellipse formulas. As a increases relative to b, the ellipse becomes more elongated and the foci move further from the centre. You can use this calculator for circles by entering a = b.

What is the semi-latus rectum?+

The semi-latus rectum l = b²/a is the half-length of the chord through one focus perpendicular to the major axis. In orbital mechanics, the semi-latus rectum determines the orbit shape: the distance r from the focus at angle θ is r = l/(1 + e cos θ), where e is eccentricity. For Earth's orbit: l = b²/a ≈ 149.6 million km (approximately equal to a since Earth's orbit is nearly circular).

How is the ellipse equation written in standard form?+

The standard form is x²/a² + y²/b² = 1, centred at the origin with the major axis along the x-axis (assuming a ≥ b). Points on the ellipse satisfy this equation. The rightmost point is (a, 0), the topmost is (0, b), and the foci are at (±c, 0) where c = √(a²−b²). If the major axis is vertical, swap a and b in the denominators.

What real-world objects are ellipses?+

Planetary orbits (Kepler's first law), the reflection of a circular beam on a surface at an angle, satellite dish cross-sections, the orbit of the Moon, car headlight reflectors, certain gears and cams, the human eye pupil under certain lighting conditions, and the footprint of a flashlight beam on a tilted wall. Even the human face outline approximates an ellipse.

What does eccentricity = 0.9 look like compared to 0.1?+

At e = 0.1: b/a = √(1 − 0.01) ≈ 0.995 - the ellipse looks almost circular. At e = 0.5: b/a = √(1 − 0.25) = 0.866 - noticeably oval. At e = 0.9: b/a = √(1 − 0.81) = 0.436 - very elongated, the minor axis is less than half the major axis. At e = 0.99: b/a ≈ 0.141 - extremely narrow, resembling a thin sliver.

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📌 Quick Tips

💡A circle is a special ellipse where a = b - its eccentricity is 0.

💡As eccentricity approaches 1, the ellipse becomes more elongated (more like a line segment).

💡Ramanujan's perimeter formula (used here) is accurate to within 0.0001% for all ellipses.

💡The semi-major axis a is always the longer radius; semi-minor b is always the shorter. If you enter a < b, the calculator swaps them automatically.

💡Earth's orbit around the Sun is an ellipse with semi-major axis 149.6 million km and eccentricity 0.0167 - nearly circular.