Ellipse area, perimeter and eccentricity explained
Where the ellipse area formula comes from, why the perimeter needs an approximation, and what eccentricity tells you about the shape.
The clean area formula
The area of an ellipse is pi times a times b, where a and b are the semi major and semi minor axes. It is a small, satisfying generalization of the circle, whose area is pi times r squared: set a and b both equal to r and the two formulas become identical. One way to see why it works is to imagine stretching a circle of radius b by a factor of a over b in one direction, which scales its area by exactly that factor and turns pi times b squared into pi times a times b. Because the formula is exact, the area figure this tool reports is limited only by the six decimal places it rounds to.
Why the perimeter is hard
Unlike the area, the exact perimeter of an ellipse cannot be written with elementary functions. It is defined by a complete elliptic integral of the second kind, which is why calculators reach for approximations instead. This tool uses Ramanujan's second formula, pi times the quantity 3(a+b) minus the square root of (3a+b)(a+3b), which is remarkably accurate for ellipses that are not extremely elongated. For a = 5 and b = 3 it returns 25.526986, and its error stays vanishingly small until the shape becomes very flat.
Reading the eccentricity
Eccentricity is a single number between 0 and 1 that captures how far an ellipse departs from a circle. It is computed as the square root of one minus the square of the ratio of the minor axis to the major axis. A value of 0 means the axes are equal and the shape is a perfect circle, while values approaching 1 describe long, cigar shaped ovals. The a = 8, b = 3 example lands at 0.927025, telling you at a glance that the ellipse is strongly stretched rather than nearly round.
Where these numbers show up
Ellipses are everywhere once you look. Planetary orbits are ellipses with the Sun at one focus, and their eccentricity determines how much the distance to the Sun varies over a year. Engineers size elliptical tanks, ducts and running tracks using the same area and perimeter formulas, and designers use ellipses for logos and layouts. Having area, circumference and eccentricity together lets you move quickly from a rough measurement to a usable figure without juggling three separate tools.