Find the Equation of an Ellipse
Enter known values. Use axes directly, or derive the missing axis from eccentricity or focus distance.
Formula Used
How to Use This Calculator
- Enter the center values h and k.
- Select the known value type from the dropdown.
- Enter axes, eccentricity, or focus distance values.
- Choose orientation when eccentricity or focus distance is used.
- Press calculate to view the equation, graph, and points.
- Use CSV or PDF export to save the current result.
Example Data Table
| Mode | Center | Input Values | Orientation | Expected Equation | Eccentricity |
|---|---|---|---|---|---|
| Axes | (0, 0) | rx = 6, ry = 4 | Horizontal | x² / 36 + y² / 16 = 1 | 0.7454 |
| Major and eccentricity | (2, -1) | a = 8, e = 0.6 | Vertical | ((x - 2)² / 40.96) + ((y + 1)² / 64) = 1 | 0.6 |
| Major and focus | (-3, 2) | a = 10, c = 6 | Horizontal | ((x + 3)² / 100) + ((y - 2)² / 64) = 1 | 0.6 |
Understanding Ellipse Equations
An ellipse is a smooth closed curve built around two axes. The longer axis is the major axis. The shorter axis is the minor axis. This calculator helps turn practical measurements into a standard equation. It also estimates eccentricity, foci, vertices, co-vertices, area, perimeter, and directrices.
Why Eccentricity Matters
Eccentricity explains how stretched an ellipse is. A value near zero means the shape is close to a circle. A value near one means the ellipse is long and narrow. In many math and science problems, eccentricity gives faster insight than axis lengths alone. It also helps compare different ellipses on the same scale.
Using Center and Axes
The standard ellipse equation uses a center point and two semi-axis values. The horizontal form places the larger denominator under the x term. The vertical form places it under the y term. This tool detects orientation when requested. It also accepts eccentricity with one semi-axis, then derives the missing axis.
Graph and Export Benefits
A graph makes the equation easier to check. You can see the center, shape, and axis direction. The table shows example inputs, so beginners can compare their results. Export buttons save the current solution as CSV or PDF. This is useful for worksheets, reports, and class notes.
Accuracy Tips
Use positive values for axis lengths. Keep eccentricity between zero and one for true ellipses. Round only after the final answer when accuracy is important. If data comes from a drawing, measure from the center to the edge, not across the full width. Always confirm whether the major axis is horizontal or vertical before using the equation.
Common Applications
Ellipse equations appear in coordinate geometry, orbital models, architecture, lens design, and engineering layouts. They are also common in exam questions because one set of inputs can produce many connected results. By showing formulas and steps, this calculator supports learning instead of only returning final numbers. It is designed for quick checks and deeper study. You can change inputs, resubmit, and compare each graph immediately. This makes patterns easier to understand. The same approach also helps when verifying answers from textbooks, assignments, or plotted design sketches before submission.
FAQs
1. What is the standard equation of an ellipse?
The standard equation is a sum of two squared coordinate terms equal to one. The denominators are the squared semi-axis lengths. The center shifts the x and y terms.
2. What does eccentricity mean?
Eccentricity measures how stretched an ellipse is. A value near zero is almost circular. A value closer to one is more elongated.
3. Can eccentricity be greater than one?
Not for an ellipse. Ellipse eccentricity is at least zero and less than one. Values greater than one describe other conic shapes.
4. What is the focus distance?
Focus distance is the distance from the center to each focus. It is called c and follows the relation c² = a² - b².
5. How do I know the orientation?
If the wider direction is left to right, the ellipse is horizontal. If the taller direction is up and down, it is vertical.
6. What are vertices?
Vertices are the two endpoints on the major axis. They sit farthest from the center along the longest direction of the ellipse.
7. What are co-vertices?
Co-vertices are endpoints on the minor axis. They show the shorter direction and help form the complete shape of the ellipse.
8. Is the perimeter exact?
The perimeter shown is an approximation. Ellipse perimeter has no simple elementary exact formula, so this tool uses Ramanujan’s accurate approximation.