Ellipse Calculator

An ellipse is defined by its semi-major axis (a) and semi-minor axis (b), along with its center coordinates (c1, c2). The area is calculated using the classic geometric formula Area = π × a × b. For the perimeter, the calculator employs Ramanujan's highly accurate approximation formula, which accounts for the ratio of the axes. Eccentricity (e), which measures how much the conic section deviates from being circular, is derived from the formula e = √(max(a,b)² - min(a,b)²) / max(a,b). Finally, the foci positions are determined by shifting along the major axis by a focal distance equal to √(|a² - b²|).

Let's find the geometric properties of an ellipse centered at the origin (0, 0) with a semi-major axis (a) of 5 units and a semi-minor axis (b) of 3 units. First, calculate the area: Area = π × 5 × 3 ≈ 47.123 square units. Next, determine the eccentricity: e = √(5² - 3²) / 5 = √(25 - 9) / 5 = √(16) / 5 = 4 / 5 = 0.8. Using Ramanujan's approximation, the perimeter evaluates to approximately 25.528 units. Finally, the foci are located along the horizontal axis at (±4, 0) because the focal distance is √(5² - 3²) = 4.

Always double-check which axis is larger before calculating foci; if the vertical semi-minor axis is actually larger than the horizontal axis, the orientation of the ellipse shifts and the foci will lie along the vertical line. When computing approximations for the perimeter, keep in mind that Ramanujan's formula becomes increasingly precise as the values of a and b approach each other, though it remains exceptionally accurate even for elongated shapes.