Ellipse Calculator
Calculate area, circumference, axes, eccentricity, and other properties of any ellipse instantly
Calculation Results
Enter ellipse measurements to calculate all properties.
Semi-Major Axis (a): units
Semi-Minor Axis (b): units
Area (A): square units
Circumference (C): units
Eccentricity (e):
Focal Distance (c): units
Linear Eccentricity: units
Flattening (f):
Ellipse Diagram
The Ellipse: A Special Oval Shape
An ellipse is a closed curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. It's a stretched circle with two distinct axes.
Key Properties of an Ellipse
- Semi-Major Axis (a) - The longest radius from center to edge.
- Semi-Minor Axis (b) - The shortest radius from center to edge.
- Focal Distance (c) - Distance from center to each focus (c² = a² - b²).
- Eccentricity (e) - Measures how much the ellipse deviates from being circular (0 = circle, 1 = parabola).
- Area (A) - The space enclosed by the ellipse (πab).
- Circumference (C) - The perimeter of the ellipse (approximation needed).
Mathematical Equations of an Ellipse
- Area: A = πab
- Circumference Approximation: C ≈ π[3(a+b) - √((3a+b)(a+3b))]
- Eccentricity: e = √(1 - (b²/a²))
- Focal Distance: c = √(a² - b²)
- Flattening: f = (a-b)/a
Why Are Ellipses Important?
- Astronomy - Planetary orbits follow elliptical paths (Kepler's 1st Law).
- Engineering - Used in arch structures, gears, and optical systems.
- Physics - Describes particle motion in potentials and wave fronts.
- Art & Design - Common in aesthetic designs and architecture.
Facts About Ellipses
- A circle is a special case of an ellipse where both axes are equal.
- The reflective property: Any ray from one focus reflects to the other focus.
- The term "ellipse" comes from the Greek word "elleipsis" meaning "falling short".
Ellipses are fundamental geometric shapes with unique mathematical properties and wide-ranging applications in science and engineering.
Major Axis (2a)
Minor Axis (2b)
Foci (F₁, F₂)
Focal Distance (c)
Area (A)
Circumference (C)
Ellipse Formulas
- Area: A = πab
- Circumference Approximation: C ≈ π[3(a+b) - √((3a+b)(a+3b))]
- Eccentricity: e = √(1 - (b²/a²))
- Focal Distance: c = √(a² - b²)
- Flattening: f = (a-b)/a
- Standard Equation: (x²/a²) + (y²/b²) = 1
Where π (pi) ≈ 3.141592653589793
How to Use the Ellipse Calculator
The Ellipse Calculator helps compute various properties based on input values. Here's how to use it:
1 Select What You Know
Choose the known values combination:
- Semi-Axes (a & b) - The two principal radii
- Area & Circumference
- Axis & Eccentricity
- Foci & Semi-Major Axis
2 Enter the Values
Type in positive numeric values for your selected parameters.
Example: a = 5, b = 3
3 Click "Calculate"
The calculator will compute:
- Both semi-axes
- Focal distance
- Eccentricity
- Area
- Circumference
- Flattening
Example Use Cases
- Calculate the area of an elliptical garden bed.
- Determine the eccentricity of a planetary orbit.
- Find the perimeter of an elliptical running track.
Example Calculations
-
Example 1: a = 5 units, b = 3 units
- Area ≈ 47.12 units²
- Circumference ≈ 25.53 units
- Eccentricity ≈ 0.8
- Focal Distance ≈ 4 units
-
Example 2: a = 10 cm, e = 0.6
- b ≈ 8 cm
- Area ≈ 251.33 cm²
- Focal Distance ≈ 6 cm
Frequently Asked Questions
An ellipse is a mathematically defined curve with two axes of symmetry and specific geometric properties. An oval is a more general term for any egg-shaped or elongated circle.
Eccentricity (e) measures how much an ellipse deviates from being circular:
- e = 0 → Perfect circle
- 0 < e < 1 → Ellipse
- e = 1 → Parabola
- e > 1 → Hyperbola
The exact calculation of an ellipse's circumference requires evaluating an elliptic integral, which cannot be expressed in elementary functions. Our calculator uses a highly accurate approximation.
- Planetary orbits (Kepler's 1st Law)
- Elliptical gears in machinery
- Whispering galleries (sound reflects between foci)
- Satellite dish shapes