Calculate Ellipse

Ellipse Formulas:

\[ Area = \pi \times a \times b \] \[ Perimeter \approx \pi \times (a + b) \times \left(1 + \frac{3h}{10 + \sqrt{4 - 3h}}\right) \] \[ \text{where } h = \frac{(a - b)^2}{(a + b)^2} \]

Area:

Perimeter:

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1. What is an Ellipse?

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 generalization of a circle.

2. How Does the Calculator Work?

The calculator uses these ellipse formulas:

\[ Area = \pi \times a \times b \] \[ Perimeter \approx \pi \times (a + b) \times \left(1 + \frac{3h}{10 + \sqrt{4 - 3h}}\right) \] \[ \text{where } h = \frac{(a - b)^2}{(a + b)^2} \]

Where:

\( a \) — Semi-major axis (longest radius)
\( b \) — Semi-minor axis (shortest radius)
\( \pi \) — Approximately 3.14159

Explanation: The area formula is exact, while the perimeter formula is a highly accurate approximation by Ramanujan.

3. Importance of Ellipse Calculations

Details: Ellipse calculations are important in astronomy (planetary orbits), engineering (elliptical gears), architecture (elliptical domes), and many other fields.

4. Using the Calculator

Tips: Enter the semi-major axis (a) and semi-minor axis (b) in meters. Both values must be positive numbers. The calculator will compute both the exact area and approximate perimeter.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a circle and an ellipse?
A: A circle is a special case of an ellipse where both axes are equal (a = b).

Q2: Why is the perimeter formula an approximation?
A: The exact calculation of ellipse perimeter requires an infinite series (elliptic integral). Ramanujan's approximation is accurate to within 0.01% for most practical purposes.

Q3: Which is the semi-major axis?
A: The semi-major axis (a) is always the longer of the two axes. If you enter them in reverse order, the calculator will still give correct results.

Q4: What are some real-world examples of ellipses?
A: Planetary orbits, racetracks, swimming pools, satellite dishes, and many architectural designs use elliptical shapes.

Q5: Can I calculate a sector of an ellipse?
A: This calculator computes the complete ellipse. Sector calculations require more complex formulas.