10
units
6
units
188.4956
sq units
51.054
units
0.8
8
units
16
units
1.666667
0.4
10
units
6
units
188.4956
sq units
51.054
units
0.8
8
units
16
units
1.666667
0.4
An ellipse is a closed curve in the plane defined as the set of all points whose sum of distances from two fixed points (the foci) is constant. It is one of the four conic sections—along with the circle, parabola, and hyperbola—and arguably the most important curve in physics and astronomy. Every planet in our solar system orbits the Sun along an elliptical path, as established by Johannes Kepler in his first law of planetary motion (1609).
The Ellipse Calculator computes the area, approximate circumference, eccentricity, focal distance, and distance between foci of an ellipse given its semi-major axis a (the longest radius) and semi-minor axis b (the shortest radius). When a = b, the ellipse reduces to a circle of radius a.
The area of an ellipse is given by the beautifully simple formula A = πab, a direct generalization of the circle's area formula A = πr². This was first proven by Archimedes using his method of exhaustion. The circumference, however, has no closed-form expression in terms of elementary functions—it requires an elliptic integral. This calculator uses Ramanujan's remarkably accurate approximation: C ≈ π[3(a+b) − √((3a+b)(a+3b))], which is accurate to within 0.01% for most practical eccentricities.
The eccentricity e of an ellipse measures how elongated it is, ranging from 0 (a perfect circle) to values approaching 1 (an extremely elongated ellipse). It is defined as e = c/a, where c = √(a² − b²) is the focal distance—the distance from the center to each focus. Earth's orbit has eccentricity ≈ 0.0167, making it nearly circular, while Pluto's orbit has eccentricity ≈ 0.2488, making it noticeably elliptical.
Ellipses have remarkable reflective properties: any ray emanating from one focus reflects off the ellipse and passes through the other focus. This property is exploited in whispering galleries (such as St. Paul's Cathedral in London and the U.S. Capitol building), where a whisper at one focus can be heard clearly at the other focus across a large room. It is also the principle behind lithotripsy, a medical procedure that uses focused shock waves to break kidney stones.
In engineering, elliptical shapes appear in satellite orbits, cam profiles, arch bridges, and optical systems. The elliptical cross-section is used in structural tubes because it offers favorable aerodynamic and bending properties. Elliptical gears produce variable-speed rotation from constant-speed input, useful in printing presses and textile machinery.
The mathematics of ellipses connects to some of the deepest areas of number theory and analysis through elliptic integrals and elliptic functions, which despite their name, extend far beyond the geometry of ellipses to encompass pendulum motion, cryptography (elliptic curve cryptography), and the proof of Fermat's Last Theorem by Andrew Wiles.
Given semi-major axis a and semi-minor axis b (with a ≥ b), the calculator computes:
$$A = \pi a b$$
$$C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]$$
$$c = \sqrt{a^2 - b^2}$$
$$e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}$$
The area formula is exact. The circumference formula is Ramanujan's first approximation (1914), which has a relative error less than h³/2 where h = ((a−b)/(a+b))². For most practical ellipses (e < 0.95), this error is negligible. The eccentricity e ranges from 0 (circle) to just below 1 (highly elongated).
The area gives the total region enclosed by the ellipse. The circumference (Ramanujan approximation) gives the total arc length of the boundary—note this is an approximation since the exact circumference requires an elliptic integral. The eccentricity quantifies the shape: 0 is a circle, values near 1 are very elongated. The focal distance c is the distance from the center to each focus, and the distance between foci is 2c. The foci lie on the major axis, symmetrically placed about the center.
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This moderately eccentric ellipse (e = 0.8) has area ≈ 188.50 sq units and circumference ≈ 51.05 units.
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With e ≈ 0.28, this ellipse is nearly circular. Its area (75.40) is close to π(5)² ≈ 78.54 for a circle of radius 5.
The semi-major axis a is the longest radius of the ellipse, measured from the center to the farthest point on the curve. The semi-minor axis b is the shortest radius, measured perpendicular to the major axis. Together they define the ellipse's size and shape. When a = b, the ellipse is a circle.
Unlike the area (πab), the circumference of an ellipse cannot be expressed using elementary functions (polynomials, roots, trig functions). It requires a complete elliptic integral of the second kind: C = 4a∫√(1−e²sin²θ)dθ. Ramanujan's approximation provides excellent accuracy without needing numerical integration.
Eccentricity e ranges from 0 to just below 1 for ellipses. At e = 0, the ellipse is a perfect circle. As e increases toward 1, the ellipse becomes increasingly elongated. Earth's orbital eccentricity is 0.017 (nearly circular), Mars is 0.093, and many comets have eccentricities above 0.9 (very elongated orbits).
A ray of light (or sound) emanating from one focus of an ellipse will reflect off the boundary and pass through the other focus. This property is used in whispering galleries, satellite dish receivers, and medical lithotripsy (using focused sound waves to break kidney stones). It follows from the equal-angle reflection law applied to the ellipse's tangent line.
Ramanujan's first approximation, C ≈ π[3(a+b) − √((3a+b)(a+3b))], has relative error less than 5×10⁻⁶ for e < 0.5 and less than 0.04% even for e = 0.95. For most engineering and scientific applications, this accuracy is more than sufficient. For extreme eccentricities (e > 0.99), numerical integration of the elliptic integral is recommended.
By convention, a is the semi-major axis (larger) and b is the semi-minor axis (smaller). If you enter a < b, the calculator still produces correct area and circumference since these formulas are symmetric in the product ab. However, eccentricity and focal distance calculations assume a ≥ b; if a < b, simply swap the values for correct interpretation.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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