Free online graphing tool
Ellipse Graph Calculator — Plot x²/a² + y²/b² = 1
An ellipse is a stretched circle: the set of points whose distances from two fixed foci add to a constant. Planetary orbits, whispering galleries and the cross-section of a rugby ball are all ellipses.
The standard equation x²/a² + y²/b² = 1 is implicit, so plot it as two halves — y = ±b√(1 − x²/a²). Both are loaded below with sliders for the semi-axes a and b.
Key points of y = b sqrt(1 - x^2/a^2) inside the visible window (-9 ≤ x ≤ 9). Pan or zoom to search a different range.
Roots (x-intercepts)
No root in this window
y-intercept
- y = 3
Turning points
- Max (3.73e-8, 3)
Intersections
No crossing with another curve
Enter a value of x
This curve is undefined somewhere in the window — gaps in the line are intentional.
What is a ellipse graph calculator?
An ellipse graph calculator plots x²/a² + y²/b² = 1, where a is the semi-major axis and b the semi-minor axis. Because the equation is implicit, plot it as two functions: y = b·√(1 − x²/a²) for the top half and its negative for the bottom. When a = b the ellipse becomes a circle.
Key facts at a glance
| Standard equation | x²/a² + y²/b² = 1 |
|---|---|
| Semi-major axis | The larger of a and b |
| Vertices | (±a, 0) and (0, ±b) |
| Foci (when a > b) | (±c, 0) with c = √(a² − b²) |
| Eccentricity | e = c/a, always between 0 and 1 |
| Circle case | a = b gives eccentricity 0 — a circle |
| Area | πab |
| Defining property | The two distances to the foci always add to 2a |
How to plot ellipse graphs — step by step
- 1
Load both halves
The graph already shows y = b sqrt(1 - x^2/a^2) and its negative — together they form the complete ellipse.
- 2
Change the axes
Drag the a slider to stretch the ellipse horizontally and b to stretch it vertically.
- 3
Make it a circle
Set a and b to the same value. The eccentricity drops to zero and the ellipse becomes a circle.
- 4
Fix the aspect ratio
Press Equal-scale so the proportions on screen match the real proportions of the curve.
Ellipse examples you can plot right now
Copy any equation into the calculator above, or open it in the full calculator with one click.
Wide ellipse
y = 3 sqrt(1 - x^2/25)
Semi-axes 5 and 3; add the negative half to complete it.
Plot this →Circle special case
y = 4 sqrt(1 - x^2/16)
a = b = 4, so this is the top half of a circle of radius 4.
Plot this →Polar orbit form
r = 4/(1 + 0.5cos(θ))
The polar equation of a conic with eccentricity 0.5 — an ellipse with a focus at the origin.
Plot this →Understanding the ellipse graph
- Eccentricity as a shape number
- Eccentricity measures how squashed the ellipse is. Zero gives a circle; values approaching 1 give an ever more elongated shape. Earth's orbit has an eccentricity of about 0.017 — very nearly circular.
- Why orbits are ellipses
- Kepler's first law states that a planet orbits with the Sun at one focus. The polar form r = l/(1 + e·cos θ) is exactly the equation an orbit follows.
- Foci and the string construction
- Pin a loop of string at two points and trace it taut: the distances to the two pins always add to the same total, which is the geometric definition of an ellipse.
Ellipse graph — frequently asked questions
How do I graph an ellipse online?
Split it into two halves. For x²/a² + y²/b² = 1, plot y = b sqrt(1 - x^2/a^2) and y = -b sqrt(1 - x^2/a^2). Both are pre-loaded above with sliders for a and b.
Why can I not type x^2/25 + y^2/9 = 1 directly?
It is an implicit equation, and this calculator plots explicit forms. Rearranging into the ± square-root halves gives exactly the same curve, and shows clearly why an ellipse is not a function.
How do I find the foci of an ellipse?
When a > b, the foci sit at (±c, 0) with c = √(a² − b²). For a = 5 and b = 3, c = 4, so the foci are at (±4, 0).
What is the eccentricity of an ellipse?
e = c/a, a number between 0 and 1 that describes how elongated the curve is. e = 0 is a circle, while values close to 1 give a long thin ellipse.
How is an ellipse different from a circle?
A circle is the special case where the two semi-axes are equal. Set a = b on the sliders above and the ellipse closes up into a circle.
How do I plot an ellipse centred somewhere else?
Replace x with (x − h) and add k outside: y = k + b sqrt(1 - (x - h)^2/a^2), plus the matching negative half.
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