The center of ellipse calculator helps you find the center of an ellipse when the equation, vertices, co-vertices, or foci are known. This enables you to determine the center of an ellipse when only a few key points are known or when the equation of the ellipse is known. Read on to learn more about the formulas to find the center based on the given data!

Where is the center of an ellipse?

The center of an ellipse is the midpoint of both the major and minor axes. These axes intersect at right angles at the center. So from the center, the distances to the vertices (located along the major axis) are equal, and the distances to the co-vertices (located along the minor axis) are also equal. The center is also equidistant from the foci of the ellipse.

How do you find the center of an ellipse in general form?

To calculate the center of an ellipse given its equation in general form, we'd need to do the following:

  1. Compare the equation to the general form given below:

    (x  c1)2a2+(y  c2)2b2=1\frac{\left(x\ -\ c_{1}\right)^{2}}{a^{2}}+\frac{\left(y\ -\ c_{2}\right)^{2}}{b^{2}}=1

  2. Identify the values of c1c_{1} and c2c_{2}.

  3. The center will be the point (c1, c2)(c_{1},\ c_{2})

How to find the center from the vertices or co-vertices

To calculate the center of the ellipse from the vertices or co-vertices, we'd need to do the following:

  1. Get the coordinates of the 2 vertices.

  2. Find the midpoint of the pair of vertices by using the midpoint calculator, which is based on the following formula:

    (x1+x22, y1+y22)\left(\frac{x_{1}+x_{2}}{2},\ \frac{y_{1}+y_{2}}{2}\right)

    where (x1, y1)\left(x_{1},\ y_{1}\right) is the first vertex and (x2, y2)\left(x_{2},\ y_{2}\right) is the second vertex.

  3. The midpoint thus obtained will be the center of the ellipse!

  4. If you know the co-vertices instead of the vertices, you may follow the same steps above using the coordinates of the co-vertices instead!

How to use the center of ellipse calculator

To use this calculator for finding the center of the ellipse, you'd need to do the following:

  1. Choose which information you have, among the following available options:

    • Equation of the ellipse;
    • Vertices of the ellipse;
    • Co-vertices of the ellipse; or
    • Foci of the ellipse.
  2. Based on the option you chose, key in the values of variables in the equation or the coordinates of the corresponding points.

  3. Voila! Based on your inputs, the calculator will show you the coordinates of the center of the ellipse!

Other ellipse calculators

If you found this tool useful, you may also want to check out our other similar tools for calculations involving ellipse, such as:

FAQ

How do I find the center of an ellipse given the foci?

In order to know how to find the center of an ellipse given the foci, we'd follow these steps:

  1. Write the coordinates of the foci.
  2. Find the midpoint of these two points.
  3. Tada! The center of the ellipse will be this midpoint!

Since the center is equidistant from the foci, the midpoint of the line joining the foci will be the center of the ellipse.

What is the center of an ellipse with vertices at (0, 6) and (0, -6)?

(0, 0) is the center of the ellipse whose vertices are the points (0, 6) and (0, -6). This is because the center is equidistant from the vertices. Thus, we can find the midpoint of the points (0, 6) and (0, -6), which will give us (0, 0) as the center of the ellipse!

Madhu Raman
Ellipse with semiaxes, foci, center and vertices marked
I know...
the ellipse equation
(x - c₁)² / a² + (y - c₂)² / b² = 1
a
b
c₁
c₂
Center
x-coordinate
y-coordinate
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