Ellipse Calculator
Calculate the area, perimeter, and eccentricity of an ellipse from its semi-major and semi-minor axes. Ideal for geometry, astronomy (orbital shapes), and engineering applications involving oval cross-sections.
About this calculator
An ellipse is defined by two axes: the semi-major axis a (the longer half) and the semi-minor axis b (the shorter half). Its area is exact and simple: A = π × a × b. The eccentricity e = √(1 − b²/a²) describes how elongated the ellipse is — 0 means a perfect circle, approaching 1 means very elongated. The perimeter has no closed-form exact solution, so approximations are used. Ramanujan's first approximation P ≈ π × [3(a + b) − √((3a + b)(a + 3b))] is popular and accurate to within 0.0004% for most ellipses. Eccentricity is fundamental in orbital mechanics, where planetary paths follow elliptical trajectories governed by Kepler's first law.
How to use
Suppose a = 7 units (semi-major axis) and b = 4 units (semi-minor axis). Area: A = π × 7 × 4 = π × 28 ≈ 87.96 square units. Eccentricity: e = √(1 − 4²/7²) = √(1 − 16/49) = √(33/49) ≈ 0.8198. For the perimeter using Ramanujan's approximation: h = (a − b)²/(a + b)² = 9/121 ≈ 0.0744; P ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))) ≈ 35.89 units. Enter a = 7 and b = 4 to verify all outputs.
Frequently asked questions
How do you calculate the area of an ellipse given semi-major and semi-minor axes?
The area formula is A = π × a × b, where a is the semi-major axis and b is the semi-minor axis. This is a direct generalization of the circle area formula A = π × r², since a circle is just an ellipse with a = b = r. The formula is exact — no approximation is needed. For example, an ellipse with a = 10 cm and b = 6 cm has area A = π × 10 × 6 ≈ 188.50 cm².
Why is there no exact formula for the perimeter of an ellipse?
Unlike area, the perimeter of an ellipse involves an elliptic integral that cannot be expressed in terms of elementary functions. Mathematicians have developed several approximations of increasing accuracy; Ramanujan's approximation is widely used because it stays within 0.0004% for most practical ellipses. The simplest approximation P ≈ π(a + b) is only accurate for near-circular ellipses. More elongated ellipses (higher eccentricity) require more refined formulas to maintain accuracy.
What does eccentricity tell you about the shape of an ellipse?
Eccentricity e = √(1 − b²/a²) is a dimensionless number between 0 and 1 that quantifies how 'stretched' an ellipse is. An eccentricity of 0 means the shape is a perfect circle (a = b). As e approaches 1, the ellipse becomes increasingly elongated and needle-like. Earth's orbit has e ≈ 0.017 (nearly circular), while Halley's Comet has e ≈ 0.967 (very elongated). Eccentricity is essential in orbital mechanics, optics, and structural engineering.