Ellipse Calculator
Calculate the area, perimeter, eccentricity, or focal distance of an ellipse from its semi-major and semi-minor axes. Useful in astronomy, engineering, and any field involving oval shapes.
About this calculator
An ellipse is defined by its semi-major axis a (the longer half-axis) and semi-minor axis b (the shorter half-axis). Its area is exact: A = πab. Unlike circles, the ellipse perimeter has no simple closed form; Ramanujan's highly accurate approximation is P ≈ π[3(a+b) − √((3a+b)(a+3b))]. Eccentricity measures how elongated the ellipse is: e = √(1 − b²/a²), ranging from 0 (circle) to near 1 (very flat). The focal distance c, the distance from the center to each focus, is c = √(a² − b²). These properties underpin planetary orbit calculations, optics, and architectural design.
How to use
Take an ellipse with semi-major axis a = 5 and semi-minor axis b = 3. Area: A = π × 5 × 3 = 15π ≈ 47.12 square units. Perimeter (Ramanujan): P ≈ π × [3(5+3) − √((3×5+3)(5+3×3))] = π × [24 − √(18 × 14)] = π × [24 − √252] ≈ π × [24 − 15.87] ≈ π × 8.13 ≈ 25.53 units. Eccentricity: e = √(1 − 9/25) = √(16/25) = 0.8. Focal distance: c = √(25 − 9) = √16 = 4 units. Enter a = 5, b = 3, select your desired property, and read off the result.
Frequently asked questions
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 that describes how much an ellipse deviates from a perfect circle. When e = 0, a equals b and the shape is a circle. As e approaches 1, the ellipse becomes increasingly elongated and flat. Earth's orbital eccentricity is about 0.017, making it nearly circular, while Halley's Comet has e ≈ 0.967, giving it a very elongated path.
Why is there no exact formula for the perimeter of an ellipse?
The arc length of an ellipse involves an elliptic integral — a type of integral with no closed-form solution in terms of elementary functions. Unlike a circle where every point is equidistant from the center, an ellipse has varying curvature, making the perimeter calculation fundamentally more complex. Mathematicians like Ramanujan and others have developed highly accurate polynomial approximations; Ramanujan's formula, used here, is accurate to within a fraction of a percent for most practical ellipses.
How is the ellipse area formula derived and why is it similar to a circle's area?
The area formula A = πab is derived by stretching a circle of radius a along one axis by a factor of b/a, which scales the area by that same factor: πa² × (b/a) = πab. This can also be proven via integration. When a = b = r, the formula reduces to πr², the familiar circle area. The formula is exact, unlike the perimeter, because area scales simply with linear dimensions.