Ellipse Calculator

Find the area, circumference, foci distance, eccentricity, vertices, and standard form equation of an ellipse using the calculator below.

Origin (h, k):
(
,
)

Properties of the Ellipse:

Semi-Major Axis:
 
Semi-Minor Axis:
 
Area (A):
 
Circumference (C):
 
Eccentricity (e):
 
Foci Distance (c):
 

Standard Form Equation:

 

Graph Coordinates

First Focus:
 
Second Focus:
 
First Vertex:
 
Second Vertex:
 
First Co-Vertex:
 
Second Co-Vertex:
 
Learn how we calculated this below


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An ellipse is a two-dimensional regular oval whose curve consists of points in which the sum of the distances from both foci are the same for any point in the curve.

diagram of an ellipse showing axis a, axis b, and foci distance c

While an ellipse is oval in shape, it is different from an oval in that it is mathematically defined, while any curve resembling a squashed circle is an oval.

An ellipse is defined by the distance from its center to the horizontal and vertical vertices. The vertices lie on the semi-major and semi-minor axes that bisect the ellipse horizontally and vertically.

The first and second vertices are the points that terminate the semi-major axis, and the first and second co-vertices are the points that terminate the semi-minor axis.

Ellipse Equation

Using the semi-major axis a and semi-minor axis b, the standard form equation for an ellipse centered at origin (0, 0) is:

x2 / a2 + y2 / b2 = 1

Where:
a = distance from the center to the ellipse’s horizontal vertex
b = distance from the center to the ellipse’s vertical vertex
(x, y) = any point on the circumference

When an ellipse is oriented horizontally, then the value of a is greater than b, and when it is oriented vertically, then the value of a is less than b.

For an ellipse that is centered at an origin other than (0, 0), the standard form equation for the ellipse is:

(x – h)2 / a2 + (y – k)2 / b2 = 1

Where:
a = distance from the center to the ellipse’s horizontal vertex
b = distance from the center to the ellipse’s vertical vertex
(h, k) = center of the ellipse
(x, y) = any point on the circumference

How to Find the Area of an Ellipse

Area is the space occupied by a two-dimensional geometric shape. The area of an ellipse can be found using a simple formula.

A = πab

Thus, the area A is equal to pi times the semi-major axis a times the semi-minor axis b.

This is very similar to the formula to find the area of a circle, but instead of multiplying a single radius by itself, the product of both axes is used.

Graphic showing the ellipse area formula where the area is equal to pi times axis a times axis b

How to Find the Circumference of an Ellipse

The circumference is the distance around a two-dimensional geometric shape, sometimes referred to as the perimeter.

Interestingly, the circumference of an ellipse cannot be exactly defined by simple mathematics, but several formulas can approximate it with varying degrees of accuracy.

A relatively simple and fairly precise formula to find the circumference of an ellipse is:

C = π( 3(a + b) – (3a + b)(a + 3b) )

The circumference C is approximately equal to pi times 3 times the semi-major axis a plus the semi-minor axis b, minus the square root of 3 times a plus b, times a plus 3 times b.

Try our circumference calculator to find the circumference of a circle.

How to Find the Foci Distance

An ellipse has two focus points, pluralized foci. The distance from the center point of the ellipse to each focus is called the foci distance.

The formula to find the foci distance for an ellipse is:

c = a² – b²

The foci distance c is equal to the square root of the semi-major axis a squared minus the semi-minor axis b squared.

How to Find the Eccentricity of an Ellipse

The eccentricity of an ellipse is the ratio of the foci distance to the semi-major axis length. The smaller the eccentricity, the closer the ellipse is to becoming a circle.

The formula to find ellipse eccentricity is:

e = c / a

The eccentricity e is equal to the foci distance c divided by the semi-major axis length a.

How to Graph an Ellipse

You can graph an ellipse by plotting its major properties on the graph, then drawing a line where the sum of the foci distances are all equal. For an ellipse centered at the origin (h, k), you can plot the foci and vertices at the following points:

  • first focus: (h – c, k)
  • second focus: (h + c, k)
  • first vertex: (h – a, k)
  • second vertex: (h + a, k)
  • first co-vertex: (h, k + b)
  • second co-vertex: (h, k – b)

For a horizontal ellipse (a > b), you can plot any point on the circumference of the ellipse using the standard form equation above.

(x – h)2 / a2 + (y – k)2 / b2 = 1

For a vertical ellipse (a < b), swap a and b in the standard form equation.

(x – h)2 / b2 + (y – k)2 / a2 = 1

To plot the line, substitute any x value less than a to find the corresponding y coordinate and any y value less than b to find the corresponding x coordinate.

You might also be interested in trying our circle calculator.