How to Calculate Ellipse
What is Ellipse?
An ellipse is an oval curve defined by two focal points. It has two axes: the semi-major axis (a, longer) and semi-minor axis (b, shorter). Ellipses appear in planetary orbits, optics, and engineering.
Formula
Area = πab; Perimeter ≈ π[3(a+b) − √((3a+b)(a+3b))]; e = √(1 − (b/a)²)
- a
- semi-major axis (half long axis) (length)
- b
- semi-minor axis (half short axis) (length)
- e
- eccentricity — measure of how "stretched" the ellipse is
Step-by-Step Guide
- 1Area = π × a × b
- 2Perimeter ≈ π × [3(a+b) − √((3a+b)(a+3b))] (Ramanujan)
- 3Eccentricity = √(1 − (b/a)²)
- 4A circle is an ellipse where a = b
Worked Examples
Input
a = 5, b = 3
Result
Area = π×5×3 = 47.12, Eccentricity ≈ 0.8
Input
a = 10, b = 6
Result
Area = 188.5, Perimeter ≈ 51.05
Frequently Asked Questions
What is eccentricity and what does it measure?
Eccentricity (e) measures how much the ellipse deviates from a circle. e=0 is a circle, e approaching 1 is very stretched.
How do I calculate the foci of an ellipse?
The distance from center to each focus is c = √(a² − b²). The foci lie on the major axis.
Why is the perimeter approximate?
Unlike circles, ellipse perimeter has no simple closed formula. Ramanujan's approximation is highly accurate.
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