Ellipse Properties Calculator
Find the area, perimeter, eccentricity, or focal distance of any ellipse given its semi-major and semi-minor axes. Ideal for orbital mechanics, optics, and engineering design.
About this calculator
An ellipse is defined by two axes: the semi-major axis a (longest half-width) and the semi-minor axis b (shortest half-width). Its area is exact: A = π × a × b. Eccentricity measures how 'stretched' the ellipse is: e = √(1 − b²/a²), ranging from 0 (circle) to just below 1 (very elongated). The focal distance between the two foci is 2c where c = √(a² − b²). Unlike area, the perimeter has no closed-form solution. Ramanujan's first approximation is P ≈ π × [3(a+b) − √((3a+b)(a+3b))]. His more accurate second formula uses h = (a−b)²/(a+b)² and gives P ≈ π(a+b) × [1 + 3h/(10 + √(4−3h))]. Both are widely used in science and engineering.
How to use
Take an ellipse with semi-major axis a = 10 units and semi-minor axis b = 6 units. Area: A = π × 10 × 6 = 60π ≈ 188.50 square units. Eccentricity: e = √(1 − 36/100) = √0.64 = 0.80. Focal distance: 2c = 2√(100 − 36) = 2√64 = 16 units. For the perimeter using Ramanujan's second formula: h = (10−6)²/(10+6)² = 16/256 = 0.0625; P ≈ π × 16 × [1 + 3(0.0625)/(10 + √(4 − 0.1875))] ≈ π × 16 × 1.0187 ≈ 51.19 units.
Frequently asked questions
What is the eccentricity of an ellipse and what does it tell you?
Eccentricity (e) is a dimensionless number between 0 and 1 that describes the shape of an ellipse. When e = 0, the ellipse is a perfect circle; as e approaches 1, the ellipse becomes increasingly elongated and needle-like. It is calculated as e = √(1 − b²/a²), where a is the semi-major and b is the semi-minor axis. In astronomy, Earth's orbital eccentricity is about 0.017, making its orbit nearly circular, while Halley's Comet has an eccentricity of about 0.967, producing a highly elongated orbit.
Why is there no exact formula for the perimeter of an ellipse?
The arc length of an ellipse requires evaluating an elliptic integral, which cannot be expressed using elementary functions like addition, multiplication, or standard trig functions. Mathematicians have proven this is a fundamental limitation, not merely a gap in current knowledge. Instead, several excellent approximations exist, most notably Ramanujan's formulas, which are accurate to better than one part in a million for most ellipses. For practical purposes, Ramanujan's second approximation is the standard choice in engineering and science.
How do the foci of an ellipse relate to its semi-major and semi-minor axes?
The two foci lie on the major axis, each at a distance c = √(a² − b²) from the center. A defining property of an ellipse is that for any point on the curve, the sum of distances to the two foci equals exactly 2a. This property is exploited in optics: an elliptical mirror reflects all light from one focus perfectly to the other, which is used in medical lithotripsy devices and some telescope designs. The total focal separation (the distance between the two foci) is 2c = 2√(a² − b²).