Ellipse Input Form
Example Data Table
Use these sample values to test the calculator and compare different ellipse layouts.
| Center | a | b | Orientation | c | Vertices | Foci |
|---|---|---|---|---|---|---|
| (0, 0) | 5 | 3 | Horizontal | 4 | (5, 0), (-5, 0) | (4, 0), (-4, 0) |
| (2, -1) | 6 | 4 | Vertical | 4.4721 | (2, 5), (2, -7) | (2, 3.4721), (2, -5.4721) |
| (-3, 2) | 10 | 8 | Horizontal | 6 | (7, 2), (-13, 2) | (3, 2), (-9, 2) |
Formula Used
The calculator uses the standard ellipse form. The center is written as (h, k). The semi-major axis is a. The semi-minor axis is b.
c = √(a² - b²) eccentricity = c / a area = πab major axis length = 2a minor axis length = 2b latus rectum = 2b² / a
Horizontal Ellipse
((x - h)² / a²) + ((y - k)² / b²) = 1 Vertices: (h + a, k), (h - a, k) Foci: (h + c, k), (h - c, k) Covertices: (h, k + b), (h, k - b)
Vertical Ellipse
((x - h)² / b²) + ((y - k)² / a²) = 1 Vertices: (h, k + a), (h, k - a) Foci: (h, k + c), (h, k - c) Covertices: (h + b, k), (h - b, k)
How to Use This Calculator
- Enter the ellipse center as h and k.
- Enter the semi-major axis value as a.
- Enter the semi-minor axis value as b.
- Choose whether the major axis is horizontal or vertical.
- Select the decimal precision for rounded results.
- Press the calculate button.
- Review vertices, covertices, foci, eccentricity, and equation.
- Use CSV or PDF buttons to save the result.
Understanding Ellipse Vertices and Foci
What the Calculator Finds
An ellipse is a smooth oval curve. It is defined by a center, two axes, two vertices, two covertices, and two foci. This calculator finds these important points from simple inputs. You only need the center, semi-major axis, semi-minor axis, and orientation. The tool then builds a complete coordinate summary.
Why Vertices Matter
Vertices mark the farthest points on the major axis. They show the main stretch of the ellipse. When the ellipse is horizontal, the vertices move left and right from the center. When it is vertical, they move up and down. These points help define the visible length and direction of the ellipse.
Why Foci Matter
Foci are special internal points. Their distance from the center is called c. This value depends on a and b. A larger difference between a and b makes the foci move farther from the center. If a and b are equal, the shape becomes a circle. In that case, the foci meet at the center.
Using the Results
The results can support geometry homework, graphing tasks, engineering layouts, and analytic geometry checks. The standard equation helps you draw or verify the ellipse. The eccentricity shows how stretched the ellipse is. A value near zero means the shape is close to circular. A value closer to one means the ellipse is highly elongated. The area and latus rectum add deeper measurement options for advanced work.
Accuracy Tips
Always enter a as the semi-major axis. It should be greater than or equal to b. Use the correct orientation before calculating. A horizontal ellipse places vertices on the x-direction. A vertical ellipse places vertices on the y-direction. Keep extra decimal places when checking detailed graphs.
Frequently Asked Questions
1. What does this ellipse calculator do?
It calculates vertices, covertices, foci, c value, eccentricity, area, axis lengths, latus rectum, and standard equation from ellipse inputs.
2. What is the semi-major axis?
The semi-major axis is half the longest width of the ellipse. It is usually written as a in standard formulas.
3. What is the semi-minor axis?
The semi-minor axis is half the shortest width of the ellipse. It is usually written as b in standard formulas.
4. How are foci calculated?
Foci use the formula c equals square root of a squared minus b squared. The value c is measured from the center.
5. What happens when a equals b?
The ellipse becomes a circle. The c value becomes zero, so both foci sit at the same point as the center.
6. How do I choose orientation?
Choose horizontal when the ellipse is wider left to right. Choose vertical when the ellipse is taller up and down.
7. Can I download the result?
Yes. The calculator includes CSV and PDF download buttons, so you can save results for reports, notes, or later checking.
8. Why must a be greater than b?
The symbol a represents the semi-major axis. Since the major axis is longest, a must be greater than or equal to b.