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Ellipse Standard Form Calculator

Created by AbdulRafay Moeen
Reviewed by Dominik Czernia, PhD
Last updated: Jan 18, 2024


To help you calculate the standard equation of an ellipse from its vertices and co-vertices, here's our ellipse standard form calculator. It uses the ellipse standard form equation to find the center and vertices of an ellipse or acts as the calculator for writing the equation of the ellipse in standard form.

The following article will also share how to find this standard form of an ellipse from its vertices.

What is an ellipse standard form?

To calculate the standard equation of an ellipse, we first need to know what makes an ellipse. Simply speaking, when we stretch a circle in one direction to create an oval, that makes an ellipse.

Here's the standard form or equation of an ellipse with its center at (0,0) and semi-major axis on the x-axis (if a>ba > b):

  • (xc1)2a2+(yc2)2b2=1\frac{(x - c_1)^2}{a^2} + \frac{(y - c_2)^2}{b^2} = 1

And here's the standard form or equation of the same ellipse with its semi-major axis on the y-axis:

  • (xc1)2b2+(yc2)2a2=1\frac{(x - c_1)^2}{b^2} + \frac{(y - c_2)^2}{a^2} = 1

, where:

  • (x,y)(x, y) - The coordinates of an arbitrary point on the ellipse;
  • (c1,c2)(c_1 , c_2) - Coordinates of the ellipse's center;
  • aa - semi-major axis (the longest distance from the ellipse center to the point on the ellipse); and
  • bb - semi-minor axis (the shortest distance from the ellipse center to the point on the ellipse).

The following equation relates the vertices ±a, co-vertices ±b, and foci ±c:

  • c2=a2b2c^2=a^2 - b^2

When given the foci and vertices coordinates of an ellipse, we can find the standard form of the ellipse.

How to use the ellipse standard form calculator?

It is very easy to use our ellipse standard form calculator:

  1. Input the vertices and co-vertices to obtain the ellipse standard form, e.g.,
  • Horizontal axis

    • First vertex V1: (-10, 0)
    • Second vertex V2: (10, 0)
  • Vertical axis

    • First vertex V3: (0, -6)
    • Second vertex V4: (0, 6)
  1. Based on the input values:
    • The center or origin of our ellipse is (0, 0); and

    • The calculator writes the equation of the ellipse in standard form:

      x2102+y262=1\frac{x^2}{10^2} + \frac{y^2}{6^2} = 1

How to find the standard form of an ellipse

The following section explains how to find the standard form of an ellipse with an example. Let's calculate the standard form of an ellipse with vertices (0, ±8) and foci (0, ±4):

  1. Rearrange the previously mentioned formula to:

    b2=a2c2b^2 = a^2 - c^2

  2. Place the values:

    b2=8242b^2 = 8^2 - 4^2

    thus,

    b2=48b^2 = 48 or b=48b = \sqrt{48}

  3. Since our ellipse's major axis is (0, ±8), we know it is in the vertical direction.

  4. Thus, our calculated standard equation of the ellipse is:

    x2482+y282=1\frac{x^2}{{\sqrt48}^2} + \frac{y^2}{8^2} = 1

FAQ

What is the standard equation of an ellipse?

Here is the standard equation of an ellipse with its center at (0, 0) and its major axis on the x-axis:

  • x²/a² + y²/b² = 1

, where:

  • (x, y) - The coordinates of an arbitrary point in the ellipse;
  • a and b - semi-major and semi-minor axes.

What is the ellipse standard form with vertices at (±13, 0) and (0, ±12)?

The equation x²/13² + y²/12² = 1 is the ellipse standard form with vertices at (±13, 0) and (0, ±12), where:

  • (-13, 0) - First vertex on the horizontal axis;
  • (13, 0) - Second vertex on the horizontal axis;
  • (0, -12) - First co-vertex on the vertical axis;
  • (0, 12) - Second co-vertex on the vertical axis; and
  • (0, 0) - Ellipse center.
AbdulRafay Moeen
Ellipse with semiaxes, foci, center and vertices marked
Ellipse standard form
(x - c1)2+(y - c2)2= 1
a2b2
a
b
c₁
c₂
Center
x-coordinate
y-coordinate
First vertex V1 (horizontal axis)
x-coordinate
y-coordinate
Second vertex V2 (horizontal axis)
x-coordinate
y-coordinate
First vertex V3 (vertical axis)
x-coordinate
y-coordinate
Second vertex V4 (vertical axis)
x-coordinate
y-coordinate
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