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Select the ellipse equation type and enter the inputs to determine the actual ellipse equation using this calculator.
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The Ellipse Equation Calculator helps you determine the equation of an ellipse along with its key parameters. Ellipses are widely used in engineering, astronomy, and optics. By applying the standard or general form of the ellipse equation, this tool makes it easy to calculate important properties quickly, making it ideal for students and math enthusiasts.
In two-dimensional geometry, an ellipse is a set of points such that the sum of the distances from two fixed points (called foci) is constant. The diagram below illustrates the main components of an ellipse.
The foci, labeled F1 and F2, lie along the major axis. The midpoint of the line connecting them is the center "O" of the ellipse. Knowing the foci is essential for calculating other parameters like vertices, eccentricity, and area.
The major axis is the longest line segment passing through the center and connecting the two farthest points on the boundary. Inputting the ellipse parameters in the calculator gives the exact length of the major axis.
The minor axis is the shortest line passing through the center, perpendicular to the major axis. AB is the major axis, CD is the minor axis. The calculator can also determine its length based on the ellipse parameters.
The semi-major axis is half of the major axis, measured from the center to the ellipse boundary, denoted by “a”.
The semi-minor axis is half of the minor axis, measured from the center to the boundary, denoted by “b”.
The distance from the center to a focus is called the focal length, denoted by “c”. If c = 0, the ellipse becomes a circle. The calculator can quickly compute the focal points.
Vertices are the endpoints of the major axis, labeled A and B. These are used for plotting the ellipse and calculating its parameters.
The center is the midpoint of both the major and minor axes, or equivalently, the midpoint of the line connecting the two foci. Denoted as "O", it can be found using the calculator instantly.
Eccentricity measures how elongated the ellipse is. It is the ratio of the distance from the center to a focus over the distance from the center to a vertex. 
The area is the space enclosed within the ellipse. Calculated using the formula A = πab, where a and b are the semi-major and semi-minor axes.
The perimeter is the distance around the ellipse. An approximation can be used for calculation.
Here are the main formulas for ellipse calculations:
| Parameter | Formula |
| Perimeter (approx.) | $$ P \approx 2\pi \sqrt{\frac{a^2+b^2}{2}} $$ |
| Area | $$ A = \pi a b $$ |
| Eccentricity | $$ e = \frac{c}{a} = \sqrt{1-\frac{b^2}{a^2}} $$ |
| Focal length | $$ c = \sqrt{a^2-b^2} $$ |
| Latus rectum | $$ \frac{2b^2}{a} $$ |
$$ \frac{(x-c_1)^2}{a^2} + \frac{(y-c_2)^2}{b^2} = 1 $$
Where:
Input:
Output:
No. If equal, the shape is a circle.
It is formed when a plane intersects a cone at an angle not parallel to the base.
Two foci. Elliptical lenses share this property.
Between 0 and 1.
A circle has equal axes (a = b), while an ellipse has different lengths for major and minor axes.
Ellipses appear in astronomy, architecture, and engineering. The ellipse equation calculator is a practical tool for computing all properties efficiently.
Source: Wikipedia - Ellipse, Mathsisfun - Ellipse
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