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 generalized case of a circle.

2. How Does the Calculator Work?

The calculator uses the standard ellipse equation:

Where:

• \( a \) — Semi-major axis length
• \( b \) — Semi-minor axis length
• \( x, y \) — Coordinates on the ellipse

Explanation: The calculator computes key properties of the ellipse including its eccentricity, area, and approximate circumference.

3. Importance of Ellipse Calculations

Details: Ellipses are fundamental in astronomy (planetary orbits), engineering (elliptical gears), architecture (elliptical arches), and many other fields.

4. Using the Calculator

Tips: Enter the lengths of the semi-major (a) and semi-minor (b) axes. Both values must be positive numbers. The calculator will automatically determine which is the major and minor axis.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a and b?
A: 'a' is always the larger of the two (semi-major axis), while 'b' is the smaller (semi-minor axis). The calculator will handle this automatically.

Q2: What does eccentricity measure?
A: Eccentricity (0 ≤ e < 1) measures how much the ellipse deviates from being circular (e=0 is a perfect circle).

Q3: Why is the circumference approximate?
A: There's no exact closed-form formula for ellipse circumference, so we use Ramanujan's approximation which is accurate to within 0.1%.

Q4: Can I graph the ellipse with this calculator?
A: This calculator provides numerical properties. For graphing, you would need to plot points using the standard equation.

Q5: What if a = b?
A: When a = b, the ellipse becomes a circle with eccentricity 0, and the circumference formula becomes exact (2πr).