Ellipse Calculator
Calculate area, perimeter, eccentricity, foci, and more from semi-major and semi-minor axes
The longest radius of the ellipse
The shortest radius of the ellipse
0 = circle, approaching 1 = very elongated
Must be less than semi-major axis
Ellipse Properties
Semi-major (a)
Semi-minor (b)
Area
Perimeter
Eccentricity (e)
0 = circle, 1 = parabola
Focal Distance (c)
Center to focus distance
Foci Coordinates
Location of both foci
Semi-latus Rectum (l)
l = b²/a
Directrix Distance
d = a/e (from center)
Aspect Ratio (a/b)
1 = circle
Formulas Used
Area
A = πab
Perimeter (Ramanujan)
P ≈ π[3(a+b) - √((3a+b)(a+3b))]
Eccentricity
e = √(1 - b²/a²)
Focal Distance
c = √(a² - b²)
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About Ellipse Calculator
What is an Ellipse?
An ellipse is a closed curve on a plane that surrounds two focal points. It is the locus of all points where the sum of the distances from two fixed points (foci) is constant. A circle is a special case of an ellipse where both focal points coincide (making the semi-major and semi-minor axes equal).
Ellipses appear throughout nature and science—from planetary orbits to the design of whispering galleries, satellite dishes, and medical imaging equipment.
How to Use This Calculator
Basic Properties Mode
- Select Basic Properties from the mode selector
- Enter the Semi-major axis (a) - the longest radius of the ellipse
- Enter the Semi-minor axis (b) - the shortest radius of the ellipse
- The calculator instantly computes all ellipse properties
From Eccentricity Mode
- Select From Eccentricity from the mode selector
- Enter the Semi-major axis (a)
- Enter the Eccentricity (e) - a value between 0 and 1
- View calculated semi-minor axis and all other properties
From Focal Distance Mode
- Select From Focal Distance from the mode selector
- Enter the Semi-major axis (a)
- Enter the Focal distance (c) - distance from center to focus
- View calculated properties including semi-minor axis
Ellipse Formulas
Basic Ellipse Formulas
| Property | Formula | Description |
|---|---|---|
| Area | A = πab | Area enclosed by the ellipse |
| Perimeter | P ≈ π[3(a+b) - √((3a+b)(a+3b))] | Ramanujan's approximation |
| Eccentricity | e = √(1 - b²/a²) | Measure of how elongated the ellipse is (0 to 1) |
| Linear Eccentricity | c = √(a² - b²) | Distance from center to each focus |
Additional Properties
| Property | Formula | Description |
|---|---|---|
| Semi-latus Rectum | l = b²/a | Half the chord through the focus parallel to directrix |
| Directrix Distance | d = a/e | Distance from center to each directrix |
| Foci Locations | (±c, 0) | Position of the two focal points |
Understanding Ellipse Properties
Semi-major Axis (a)
The semi-major axis is the longest radius of an ellipse, extending from the center to the farthest point on the perimeter. It defines the "length" of the ellipse.
Semi-minor Axis (b)
The semi-minor axis is the shortest radius of an ellipse, extending from the center perpendicular to the major axis. It defines the "width" of the ellipse.
Eccentricity (e)
Eccentricity measures how "stretched" an ellipse is compared to a circle:
- e = 0: Perfect circle
- 0 < e < 1: Ellipse (higher values = more elongated)
- e = 1: Parabola (not an ellipse)
Foci
The two focal points are special locations inside the ellipse. Any point on the ellipse has the same total distance to both foci. This property is used in:
- Whispering galleries (sound reflects from one focus to the other)
- Lithotripsy (medical procedure using focused sound waves)
- Satellite dish design
Real-World Applications
Astronomy
- Planetary orbits are ellipses with the Sun at one focus (Kepler's First Law)
- Comets follow highly eccentric elliptical orbits
- Satellite trajectories
Architecture & Design
- Elliptical domes and arches
- Whispering galleries (St. Paul's Cathedral, US Capitol)
- Decorative patterns and logos
Engineering
- Gears with elliptical shapes
- Cam mechanisms
- Optical systems (elliptical mirrors)
Medical Technology
- Lithotripsy for kidney stone treatment
- MRI and CT scanner designs
Frequently Asked Questions
What's the difference between an ellipse and an oval?
Mathematically, an ellipse has a precise definition based on the two foci. An "oval" is a more general term for any elongated, closed curve. All ellipses are ovals, but not all ovals are ellipses.
Why is there no exact formula for ellipse perimeter?
Unlike a circle, the perimeter of an ellipse cannot be expressed as a simple closed-form formula. It requires elliptic integrals or approximations. Ramanujan's approximation used in this calculator is accurate to about 0.0005% for most ellipses.
How do I know if my ellipse values are valid?
The semi-major axis (a) must be greater than or equal to the semi-minor axis (b). Both must be positive numbers. Eccentricity must be between 0 and 1.
What happens when a = b?
When the semi-major and semi-minor axes are equal, the ellipse becomes a perfect circle. The eccentricity becomes 0, and both foci merge at the center.
How accurate is this calculator?
This calculator uses JavaScript's Math library for high precision calculations. The perimeter uses Ramanujan's first approximation, which is extremely accurate for all practical purposes.