Ellipse Area and Circumference Calculator
Calculate the area, circumference (perimeter), eccentricity, and focal distance of an ellipse using its semi-major and semi-minor axes. Essential for astronomy, engineering, and analytic geometry.
About this calculator
An ellipse is defined by its semi-major axis a (half the longest diameter) and semi-minor axis b (half the shortest diameter), where a ≥ b > 0. Its area is exact: Area = π × a × b. Unlike a circle, the ellipse circumference has no closed-form solution; two common approximations are used. Ramanujan's approximation: C ≈ π × (a + b) × (1 + 3h / (10 + √(4 − 3h))), where h = (a − b)² / (a + b)², is highly accurate for most ellipses. The simpler Ramanujan I formula is C ≈ π × (3(a + b) − √((3a + b)(a + 3b))). Eccentricity e = √(1 − b²/a²) measures how far the ellipse deviates from a circle (e = 0 is a perfect circle; e approaching 1 is very elongated). The focal distance c = √(a² − b²) gives the distance from the center to each focus.
How to use
Consider an ellipse with semi-major axis a = 5 units and semi-minor axis b = 3 units. Area = π × 5 × 3 ≈ 47.12 square units. Eccentricity: e = √(1 − 9/25) = √(16/25) = 0.8. Focal distance: c = √(25 − 9) = √16 = 4 units. For circumference using Ramanujan's method: h = (5 − 3)² / (5 + 3)² = 4/64 = 0.0625; C ≈ π × 8 × (1 + 3 × 0.0625 / (10 + √(4 − 0.1875))) ≈ π × 8 × 1.0187 ≈ 25.53 units. Enter a = 5, b = 3, and select your desired output.
Frequently asked questions
Why is there no exact formula for the circumference of an ellipse?
Unlike a circle, whose circumference is simply 2πr, the arc length of an ellipse involves an elliptic integral — a class of integrals that cannot be expressed in terms of elementary functions. Mathematicians like Ramanujan developed highly accurate polynomial approximations to make practical computation possible. Ramanujan's second approximation is accurate to within a fraction of a percent for most ellipses. For near-circular ellipses the simple approximation 2π√((a² + b²)/2) also works well.
What does the eccentricity of an ellipse tell you about its shape?
Eccentricity e = √(1 − b²/a²) is a dimensionless number between 0 and 1 for an ellipse. When e = 0, the ellipse is a perfect circle (a = b). As e approaches 1, the ellipse becomes increasingly elongated and flattened. Earth's orbit has an eccentricity of about 0.017 (nearly circular), while Halley's Comet has e ≈ 0.967 (extremely elongated). Eccentricity is fundamental in orbital mechanics, optics (elliptical mirrors), and structural engineering.
How is the focal distance of an ellipse calculated and what are foci used for?
The focal distance c is the distance from the center of the ellipse to each focus, given by c = √(a² − b²). The two foci lie on the major axis, each at distance c from the center. A defining property of an ellipse is that the sum of distances from any point on the ellipse to the two foci is constant and equal to 2a. Foci are critical in astronomy (planets orbit the Sun, which sits at one focus), in acoustics (whispering galleries), and in optics where elliptical reflectors concentrate light or sound at a focal point.