Free Ellipse Calculator

An ellipse calculator streamlines the process of extracting geometric properties from an elliptical shape. This free ellipse calculator takes the standard‑form equation of an ellipse and instantly returns the center, vertices, foci, area, perimeter, and eccentricity. It works equally well as an oval area calculator, an ellipse perimeter calculator, or an ellipse foci calculator, eliminating the need for manual formula work and letting you concentrate on the results.

What Is an Ellipse?

An ellipse is a closed curve obtained when a right circular cone is intersected by a plane at an angle to its axis. If the intersecting plane is parallel to the cone’s base, the result is a circle — a special case of an ellipse. Geometrically, an ellipse is defined as the set of all points for which the sum of the distances to two fixed points (the foci) remains constant. In a circle, both foci coincide at a single point.

Standard Equation of an Ellipse

The standard form of the ellipse equation is:

$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$

Here, $(h, k)$ are the coordinates of the center, $a$ is the semi‑major axis (distance from the center to the farthest vertex along the horizontal direction), and $b$ is the semi‑minor axis (distance to the vertical vertex). When $a>b$, the ellipse is oriented horizontally; when $b>a$, it is vertical. If $a=b$, the equation reduces to $(x-h)^2+(y-k)^2=a^2$, which is the familiar equation of a circle of radius $a$.

Area of an Ellipse

The area enclosed by an ellipse is given by the simple product of its semi‑axes multiplied by $\pi$:

$$A=\pi\,a\,b$$

For a circle ($a=b$), this becomes $A=\pi a^2$, matching the standard circle area formula. As an example, an ellipse with $a=5$ and $b=4$ has an area of $20\pi\approx 62.83$ square units.

Perimeter of an Ellipse

Determining the perimeter (circumference) of an ellipse is far more challenging than finding its area because no exact closed‑form expression exists. Instead, high‑quality approximations are used. This calculator relies on the well‑known approximation developed by Srinivasa Ramanujan, which provides excellent accuracy for most practical ellipses.

Eccentricity of an Ellipse

Eccentricity measures how much an ellipse deviates from a circle. It can be computed directly from the semi‑axes:

• For a horizontal ellipse ($a>b$): $$e=\frac{\sqrt{a^{2}-b^{2}}}{a}$$
• For a vertical ellipse ($b>a$): $$e=\frac{\sqrt{b^{2}-a^{2}}}{b}$$

A circle corresponds to $e=0$; as the ellipse becomes more elongated, $e$ approaches 1. Using the same example ($a=5$, $b=4$), the eccentricity is $e=\sqrt{25-16}/5=3/5=0.6$.

Center, Foci, and Vertices

From the standard equation, the center is directly $(h, k)$. The vertices are always located at $(h\pm a, k)$ and $(h, k\pm b)$, regardless of orientation. The foci depend on which axis is longer:

• If $a>b$ (horizontal ellipse), the foci lie on the horizontal axis:
  $(h\pm c, k)$ where $c=\sqrt{a^{2}-b^{2}}$.
• If $b>a$ (vertical ellipse), the foci lie on the vertical axis:
  $(h, k\pm c)$ where $c=\sqrt{b^{2}-a^{2}}$.

These points, together with the area, perimeter, and eccentricity, completely describe the ellipse’s geometry.

With this free ellipse calculator, you can obtain all the above parameters in seconds by simply entering the standard equation. It is a versatile tool that functions as an ellipse area calculator, ellipse perimeter calculator, ellipse eccentricity calculator, and ellipse foci calculator all in one.