Formula Used
The standard ellipse equation is x²/a² + y²/b² = 1, where a is the semi-major axis and b is the semi-minor axis.
Major axis length = 2a. Minor axis length = 2b. Focal distance c = √(a² - b²). Eccentricity e = c/a.
Area = πab. Semi-latus rectum = b²/a. The perimeter is estimated with Ramanujan's approximation.
For an orbital ellipse, a = (ra + rp) / 2. Eccentricity equals (ra - rp) / (ra + rp).
How to Use This Calculator
- Select the calculation method that matches your known data.
- Enter only the values needed for that method.
- Keep every distance in the same unit.
- Use eccentricity from 0 to less than 1.
- Choose the number of decimal places.
- Press Calculate to view the result above the form.
- Use CSV or PDF download for reports.
Example Data Table
| Method |
Known Data |
Semi-major |
Semi-minor |
Major Axis |
Minor Axis |
Eccentricity |
| Semi-axes |
a = 10, b = 6 |
10 |
6 |
20 |
12 |
0.8000 |
| Area and eccentricity |
Area = 251.3274, e = 0.6 |
10 |
8 |
20 |
16 |
0.6000 |
| Focal distance and eccentricity |
c = 4, e = 0.5 |
8 |
6.9282 |
16 |
13.8564 |
0.5000 |
| Periapsis and apoapsis |
rp = 3, ra = 9 |
6 |
5.1962 |
12 |
10.3923 |
0.5000 |
Understanding Ellipse Axis Lengths
An ellipse appears in optics, orbital physics, wave tanks, and instrument design. Its two main axes describe its widest and narrowest spans. The major axis passes through both foci. The minor axis crosses the center at a right angle. When these lengths are known, many other properties become easy to estimate.
Why Axis Length Matters
Axis length controls the shape of a projected orbit or beam spot. A long major axis can show high stretch. A short minor axis can show strong compression. In orbital work, the semi-major axis also relates to energy. In lab work, the same value helps compare lenses, tracks, and measured paths. The calculator keeps the larger value as the major direction.
Useful Input Paths
Different problems give different data. Some experiments provide two measured spans. Some astronomy tasks give periapsis and apoapsis. Some geometry tasks give area and eccentricity. A focusing setup may provide focal distance with eccentricity. Each option leads back to semi-major length a and semi-minor length b. From those two values, the full axes are twice the semi-axis lengths.
Accuracy and Assumptions
The tool assumes an ideal ellipse on a flat plane. It also assumes positive distances. Eccentricity must be less than one for an ellipse. A value near zero means the shape is close to a circle. A value near one means the ellipse is very stretched. Rounded results may hide small changes, so increase precision for careful reports.
Physics Use Cases
Use the result table to compare measured motion with a model. Check focal separation when locating sources or receivers. Review latus rectum when studying conic paths. Estimate perimeter with Ramanujan's approximation when a fast boundary length is enough. Export the result for lab notes, homework, or design records. Always keep units consistent before entering values. Mixed units will create misleading axes and areas.
Reading the Output
The major axis is the longest full width. The minor axis is the shortest full width. The focal distance is measured from the center to one focus. The focal separation is twice that distance. The area describes enclosed space. The axis ratio and flattening explain shape quality. These values help compare trials quickly. They also support clear error checks.
FAQs
What is the major axis of an ellipse?
The major axis is the longest full width of the ellipse. It passes through the center and both foci. Its length equals twice the semi-major axis.
What is the minor axis?
The minor axis is the shortest full width through the center. It is perpendicular to the major axis. Its length equals twice the semi-minor axis.
Can eccentricity be equal to one?
No. An ellipse has eccentricity from 0 to less than 1. A value of 1 describes a parabolic boundary, not an ellipse.
Why does the calculator reorder axes?
The semi-major axis must be the larger semi-axis. If inputs are reversed, the calculator reorders them to keep the ellipse formulas consistent.
Which units should I use?
Use one consistent unit for every distance input. The output axes use that same unit. Area results use the square of the chosen unit.
How is focal distance found?
Focal distance is found with c = √(a² - b²). It measures the distance from the ellipse center to one focus.
Is the perimeter exact?
The perimeter shown is an estimate using Ramanujan's approximation. It is accurate for many practical physics and geometry tasks.
Can this help with orbital paths?
Yes. Use the periapsis and apoapsis option for simple elliptical orbit calculations. It returns axis lengths, eccentricity, and related shape values.