Ellipse Graph (x²/a² + y²/b² = 1)

Enter semi-axes a and b to graph the ellipse x²/a² + y²/b² = 1. Automatically computes area, perimeter, eccentricity, and foci. When a = b the ellipse becomes a circle.

Last updated: 2026-08-26

What Is an Ellipse Graph?

This ellipse graph tool lets you plot the curve x²/a² + y²/b² = 1 simply by entering the semi-axis a along the x-direction and the semi-axis b along the y-direction. It automatically calculates the area, perimeter (approximate), eccentricity, and coordinates of the foci. When a = b, the ellipse becomes a perfect circle, so you can see directly on the graph that a circle is just a special case of an ellipse.

Unlike area, the perimeter of an ellipse has no closed-form expression in elementary functions — it can only be written using a special integral called an elliptic integral. This tool uses Ramanujan's approximation formula to compute a perimeter that is accurate enough for practical purposes. Among the conic sections (circle, ellipse, parabola, and hyperbola), the ellipse is especially useful in the real world, from planetary orbits to architectural acoustics.

How to Use the Ellipse Graph Tool

Enter the semi-axis a along x Enter the radius measured along the x-direction, whether it is the semi-major or semi-minor axis. It must be greater than 0.
Enter the semi-axis b along y Enter the radius measured along the y-direction. Set it equal to a to draw a perfect circle.
Check the graph The ellipse is drawn based on the values of a and b you entered, and the foci are marked with diamond-shaped markers.
Review area, perimeter, and eccentricity The automatically calculated area, approximate perimeter, eccentricity e, and the coordinates of the foci are all listed for you.

Tips for getting more out of it

a is the semi-axis along the x-axis, b is along the y-axis. When a = b, the ellipse is a perfect circle.
Eccentricity e measures how elongated the ellipse is. e = 0 means a perfect circle; values close to 1 produce a very elongated ellipse.
Area is computed exactly as πab. The perimeter has no closed-form in elementary functions, so this tool uses the Ramanujan approximation: π(3(a+b) − √((3a+b)(a+3b))).
Changing the display range does not change the ellipse shape. The foci (orange diamonds) are shown on the major axis.

Ways to Use the Ellipse Graph Tool

Understanding planetary and satellite orbits

By Kepler's laws, planets travel along elliptical orbits. You can change the eccentricity and see intuitively how the shape of an orbit changes.

Calculating the area and perimeter of an ellipse

When designing or crafting an elliptical part, you can calculate its area and edge length directly from the semi-axes.

Learning how eccentricity classifies shapes

You can confirm, both numerically and visually, that an eccentricity of 0 gives a perfect circle and values closer to 1 give an increasingly elongated ellipse.

Practicing high school or college conic sections

You can study the standard form of an ellipse and the relationship between its foci and eccentricity hands-on, by changing the values yourself.

Glossary

Eccentricity: A value between 0 and 1 that describes how flattened an ellipse is. A value of 0 gives a perfect circle, and values closer to 1 give a more elongated ellipse.
Focus (foci): Two special points on the major axis such that, for every point on the ellipse, the sum of the distances to the two foci is constant.
Semi-major / semi-minor axis: The distance from the center of the ellipse to its farthest point is the semi-major axis, and the distance to its nearest point is the semi-minor axis. Whether a or b is the semi-major axis depends on which value is larger.
Elliptic integral: The integral that appears when calculating the exact perimeter of an ellipse. It cannot be written as a closed-form elementary function, so this tool uses Ramanujan's approximation instead.
Circle: A special ellipse in which the semi-major and semi-minor axes are equal (a = b). Its eccentricity is 0.

FAQ

When a = b the ellipse becomes a perfect circle. Eccentricity is 0 and the two foci coincide at the centre.

The exact perimeter of an ellipse involves an elliptic integral that has no closed-form in elementary functions (except when a = b). This tool uses the Ramanujan approximation, which is accurate to within a fraction of a percent for typical shapes.

When a ≥ b the major axis is horizontal and the foci are at (±c, 0) where c = √(a²−b²). When b > a the major axis is vertical and the foci are at (0, ±c) where c = √(b²−a²). They appear as orange diamonds on the graph.

Side Note — Ellipses in planetary orbits

The word "ellipse" comes from the ancient Greek "ἔλλειψις" (elleipsis), meaning deficiency or falling short — when a cone is sliced at an angle, the cross-section falls short of a full circle.

Kepler's first law (1609) states that every planet orbits the Sun in an ellipse, with the Sun at one focus. Earth's orbital eccentricity is about 0.0167 (nearly circular), while Pluto's is about 0.249.

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