🔵 Ellipse Calculator
Calculate area, perimeter, eccentricity, and other ellipse properties
The longer radius (half the major axis)
The shorter radius (half the minor axis)
How to Use This Calculator
Enter Semi-Axes
Input the semi-major axis (a) and semi-minor axis (b). The semi-major axis should be greater than or equal to the semi-minor axis.
Click Calculate
Press the "Calculate Ellipse Properties" button to compute area, perimeter, eccentricity, and other properties.
Review Results
View all calculated ellipse properties including area, perimeter, eccentricity, and distance between foci.
Formulas
Area = πab
Where a = semi-major axis, b = semi-minor axis
Perimeter ≈ π[3(a + b) - √((3a + b)(a + 3b))]
Ramanujan's approximation (very accurate)
Eccentricity = √(1 - b²/a²)
Measures how "stretched" the ellipse is (0 = circle, close to 1 = very elongated)
Distance Between Foci = 2√(a² - b²)
The distance between the two focus points
Standard Equation:
x²/a² + y²/b² = 1
Example: Calculate ellipse properties for a = 10, b = 6
Area = π × 10 × 6 = 60π ≈ 188.50 square units
Eccentricity = √(1 - 6²/10²) = √(1 - 36/100) = √0.64 = 0.8
Distance Between Foci = 2√(10² - 6²) = 2√(100 - 36) = 2√64 = 16 units
About Ellipse Calculator
The Ellipse Calculator helps you find the area, perimeter, eccentricity, and other important properties of an ellipse. An ellipse is an oval-shaped curve defined as the set of points where the sum of distances to two foci is constant.
When to Use This Calculator
- Astronomy: Calculate planetary orbits (elliptical paths)
- Engineering: Design elliptical gears, cams, and mechanical components
- Architecture: Design elliptical windows, arches, and structures
- Geometry: Study ellipse properties and calculations
- Physics: Analyze orbital mechanics and elliptical motion
- Education: Learn and practice ellipse mathematics
Why Use Our Calculator?
- ✅ Comprehensive Properties: Calculates area, perimeter, eccentricity, and foci distance
- ✅ Instant Results: Get all ellipse properties immediately
- ✅ Accurate Perimeter: Uses Ramanujan's approximation for precise perimeter calculation
- ✅ Step-by-Step Display: See calculation formulas with your values
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Ellipses
An ellipse is a conic section (like circles, parabolas, and hyperbolas) with unique properties:
- Foci: Two fixed points inside the ellipse. Sum of distances from any point to both foci is constant (2a)
- Eccentricity: Measures how "stretched" the ellipse is. Range: 0 (circle) to 1 (highly elongated)
- Semi-Axes: a = semi-major (longer), b = semi-minor (shorter)
- Major Axis: The longest diameter (length = 2a)
- Minor Axis: The shortest diameter (length = 2b)
- Circle Special Case: When a = b, the ellipse becomes a circle (eccentricity = 0)
Real-World Applications
Planetary Orbits: Planets orbit the sun in elliptical paths with the sun at one focus. Earth's orbit has eccentricity ≈ 0.017 (nearly circular).
Satellite Communication: Many satellites follow elliptical orbits. The eccentricity determines how elongated the orbit is.
Architecture: Elliptical domes and arches use ellipse geometry for structural and aesthetic design. The Colosseum in Rome features elliptical structure.
Whispering Galleries: Elliptical rooms have special acoustic properties - sound from one focus can be clearly heard at the other focus.
Frequently Asked Questions
What is an ellipse?
An ellipse is an oval-shaped curve defined as the set of all points where the sum of distances to two fixed points (foci) is constant. It's a conic section formed by slicing a cone at an angle.
How is an ellipse different from a circle?
A circle is a special ellipse where both semi-axes are equal (a = b), giving eccentricity = 0. Ellipses have different a and b values, making them elongated. Circles have one center; ellipses have two foci.
What is eccentricity?
Eccentricity (e) measures how "stretched" an ellipse is. Range: 0 to 1. e = 0 means circle. e close to 1 means very elongated. Formula: e = √(1 - b²/a²).
Why is there no exact formula for ellipse perimeter?
The ellipse perimeter involves an elliptic integral, which cannot be expressed in elementary functions. We use approximations like Ramanujan's formula, which is extremely accurate (error < 0.01% for most ellipses).
What are the foci of an ellipse?
Foci are two fixed points inside an ellipse. For any point on the ellipse, the sum of distances to both foci equals 2a (the major axis length). The distance between foci is 2√(a² - b²).
Can b be greater than a?
By convention, a is always the semi-major axis (longer) and b is the semi-minor axis (shorter). If you have values where b > a, simply swap them - the ellipse properties remain the same.
Why are ellipses important in astronomy?
Planetary orbits are ellipses with the sun at one focus (Kepler's First Law). Elliptical orbits explain why planets are sometimes closer to the sun (perihelion) and sometimes farther (aphelion).