Ellipse Circumference Calculator
Calculate the circumference (perimeter) of an ellipse using Ramanujan's approximation formula
Calculate Ellipse Circumference
Longest radius from center to edge
Shortest radius from center to edge
Ellipse Circumference Results
Formula used: p ≈ π(a + b)[1 + 3h/(10 + √(4 - 3h))]
Where h = (a - b)²/(a + b)² = 0.000000
Eccentricity: 0.0000 (Circle)
Simple approximation: π(a + b) = 0.0000 cm (NaN% error)
Approximation Accuracy
Example Calculation
Earth's Orbit Example
Semi-major axis (a): 149.6 million km
Semi-minor axis (b): 149.6 million km (approximately circular)
Eccentricity: 0.017 (very low, nearly circular)
Step-by-step Calculation
1. Calculate h = (a - b)² / (a + b)² ≈ 0.00029
2. Calculate √(4 - 3h) ≈ 1.9996
3. Calculate 3h/(10 + √(4 - 3h)) ≈ 0.0000435
4. Circumference ≈ π(a + b)(1 + 0.0000435)
Result: ≈ 939.95 million km
Ramanujan's Formula
Where:
- • p = circumference
- • a = semi-major axis
- • b = semi-minor axis
- • h = (a - b)²/(a + b)²
Note: This approximation is accurate to within 0.5% for all ellipses, making it one of the best simple formulas available.
Quick Reference
When a = b, the ellipse becomes a circle
Eccentricity ranges from 0 (circle) to 1 (parabola)
Area = π × a × b
Simple approximation: π(a + b)
No exact formula exists for ellipse perimeter
Understanding Ellipse Circumference
What is Ellipse Circumference?
The circumference of an ellipse, also called its perimeter, is the total length of the boundary around the elliptical shape. Unlike a circle, there is no exact simple formula for the circumference of an ellipse.
Why Use Approximations?
- •The exact formula involves infinite series or elliptic integrals
- •Ramanujan's approximation is simple yet highly accurate
- •Error is typically less than 0.5% for any ellipse
- •Perfect for engineering and scientific applications
Other Approximation Methods
Simple Approximation
p ≈ π(a + b)
Quick but less accurate
Padé Approximation
p ≈ π(a + b)[1 + 3h/(10 + √(4 - 3h))]
Ramanujan's method (used here)
Infinite Series
Exact but computationally intensive
Uses elliptic integrals
Historical Note: Srinivasa Ramanujan developed this approximation in the early 1900s. It remains one of the most elegant and accurate simple formulas.