Ellipse Calculator
Calculate the area, perimeter (circumference), and eccentricity of an ellipse.
What Is the Ellipse Calculator?
The Ellipse Calculator computes the area, perimeter (circumference), eccentricity, focal distance, semi-latus rectum, and equation of an ellipse from its semi-major axis a and semi-minor axis b. It handles both horizontal (a > b) and vertical (b > a) ellipses.
Formula
How to Use
Enter the semi-major axis a (half the longer diameter) and semi-minor axis b (half the shorter diameter). The calculator returns area, perimeter (using Ramanujan's approximation), eccentricity (0 for circle, approaching 1 for very elongated ellipse), and focal distance c.
Example Calculation
Ellipse with a=5, b=3: Area = π×5×3 ≈ 47.12 sq units. c = √(25−9) = √16 = 4 (foci at ±4). Eccentricity e = 4/5 = 0.8. Perimeter ≈ π(3(8)−√((19)(11))) ≈ π(24−√209) ≈ π(24−14.46) ≈ 30.0 units.
Understanding Ellipse
An ellipse is the set of all points where the sum of distances to two fixed points (the foci) is constant. It is a conic section — the curve formed when a plane intersects a cone at an angle less than the slope of the cone's surface. Every circle is a special case of an ellipse where both foci coincide at the center (e=0).
Ellipses are fundamental in astronomy: Kepler's first law states that planets orbit the Sun in elliptical paths with the Sun at one focus. The Earth's orbit has eccentricity ≈ 0.017 (nearly circular), while highly elliptical cometary orbits have e approaching 1. Elliptical reflectors have the property that light from one focus reflects through the other — used in medical lithotripsy and whispering galleries.
In engineering and design, ellipses appear in the cross-sections of shafts and pressure vessels under combined loading, in CAD curve fitting, in the design of oval race tracks and stadiums, and in optics (ellipsoidal mirror profiles). The ellipse equation x²/a² + y²/b² = 1 is one of the most important curves in analytical geometry.
Frequently Asked Questions
What is the difference between semi-major and semi-minor axis?
The semi-major axis a is half the length of the longest diameter of the ellipse. The semi-minor axis b is half the length of the shortest diameter. When a=b, the ellipse is a circle.
Why is there no exact formula for ellipse perimeter?
The arc length integral for an ellipse involves an elliptic integral, which has no closed form in elementary functions. Ramanujan's approximation π(3(a+b)−√((3a+b)(a+3b))) is accurate to within 0.001% for most ellipses.
What is eccentricity?
Eccentricity e = c/a measures how elongated the ellipse is. e=0 for a perfect circle, 0 < e < 1 for an ellipse, e=1 for a parabola, e > 1 for a hyperbola. Earth's orbital eccentricity is about 0.0167.
What are the foci of an ellipse?
The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to both foci is constant (equals 2a). The foci are at distance c = √(a²−b²) from the center.
Is this calculator free?
Yes, completely free with no account required.
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