This all-in-one online Ellipse Calculator helps to calculate the missing parameters of an ellipse provided any two parameters of the ellipse are known.
Ellipse Equations
An ellipse is a plane curve surrounding two focal points \(F\), separated by a distance \(2c\), such that for all points on the curve, the sum of the two distances to the focal points is a positive constant \(2a\). Here \(a\) is the semimajor axis. The corresponding parameter \(b\) is known as the semiminor axis.
The equation of a standard ellipse centered at the origin of the coordinate system with width \(2a\) and height \(2b\) is:
$${\frac {{x}^{2}}{{a}^{2}}}+{\frac {{y}^{2}}{{b}^{2}}}=1.$$
Assuming \(a \gt b\), the foci are \((\pm c, 0)\) for \(c={\sqrt {a^{2}-b^{2}}}\). The standard parametric equation of ellipse is:
$$(x,y)=(a\cdot cos(t), b\cdot sin(t)), \quad 0\leq t\leq 2\pi.$$
The elongation of an ellipse is measured by its eccentricity \(e\), a number ranging from \(e=0\) (the limiting case of a circle) to \(e=1\) (the limiting case of infinite elongation which is a parabola). The eccentricity \(e\) is defined as follows:
$$e={\frac {c}{a}}=\sqrt{1-\frac{b^{2}}{a^{2}}}$$
The area of an ellipse is expressed by the following formula:
$$A=πab.$$
Note that we don’t specify units of measure in our Ellipse calculator. We assume that area is measured in square units of length.
An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution.
Ellipses are common in nature and are found in physics, astronomy and engineering.
So, the orbits of the planets in the Solar System are ellipses but the eccentricities are so small for most of the planets that they look circular at first glance. Pluto and Mercury are exceptions: their orbits are sufficiently eccentric that they can be seen by inspection to not be circles. The same applies to satellites and other spacecraft orbiting the earth.
Elliptical gears are used in engineering because the way they interact with each other creates the varying speed.
Related calculators
Check out our other geometry calculators such as Triangle Calculator or Sphere Calculator.