Ellipse Perimeter Calculator
Calculate ellipse perimeter using Ramanujan's approximation with area, eccentricity, and comparison methods
Calculate Ellipse Perimeter
Length from center to edge (any direction)
Length from center to edge (perpendicular direction)
Ellipse Perimeter Results
Formula used: p ≈ π(a + b)[1 + 3h/(10 + √(4 - 3h))]
Where h = (a - b)²/(a + b)² = 0.000000
Semi-major axis (a): 0.0000 cm
Semi-minor axis (b): 0.0000 cm
Method Comparison
π(a + b) = 0.0000 cm
Error: NaN%
0.0000 cm
Difference: 0.000000 cm
Ramanujan's Approximation Accuracy
Example Calculation
Example: Athletic Track
Semi-major axis (a): 5 units
Semi-minor axis (b): 3 units
From competitor example: Expected perimeter ≈ 25.527 units
Step-by-step Calculation
1. Calculate h = (a - b)² / (a + b)² = (5 - 3)² / (5 + 3)² = 4/64 = 0.0625
2. Calculate √(4 - 3h) = √(4 - 3×0.0625) = √3.8125 ≈ 1.9527
3. Calculate 3h/(10 + √(4 - 3h)) = 0.1875/(10 + 1.9527) ≈ 0.0157
4. Perimeter ≈ π(a + b)(1 + 0.0157) = π × 8 × 1.0157
Result: ≈ 25.527 units (matches expected!)
Ramanujan's Formula
Where:
- • p = perimeter (circumference)
- • a = semi-major axis
- • b = semi-minor axis
- • h = (a - b)²/(a + b)²
Historical Note: Developed by Srinivasa Ramanujan (~1914), this formula is remarkably accurate and elegant for practical calculations.
Related Formulas
Area
A = π × a × b
Eccentricity
e = √(1 - (b/a)²)
Standard Form
(x-h)²/a² + (y-k)²/b² = 1
Quick Reference
No exact simple formula exists for ellipse perimeter
Ramanujan's approximation: <0.5% error
When a = b, ellipse becomes a circle
Larger eccentricity = more elongated shape
Foci distance from center: c = e × a
Understanding Ellipse Perimeter
What is Ellipse Perimeter?
The perimeter (or circumference) of an ellipse is the total distance around its boundary. Unlike circles, ellipses don't have a simple exact formula for perimeter calculation, requiring sophisticated approximation methods.
Why Ramanujan's Formula?
- •Exceptional accuracy (within 0.5% of exact value)
- •Simple enough for practical calculations
- •Works for all ellipse shapes and sizes
- •More accurate than simple π(a + b) approximation
Approximation Methods Comparison
Simple Approximation
p ≈ π(a + b)
Quick but can have significant error
Ramanujan's Approximation
p ≈ π(a + b)[1 + 3h/(10 + √(4 - 3h))]
Highly accurate and practical
Infinite Series (Exact)
Uses elliptic integrals
Exact but computationally complex
Mathematical Insight: The exact perimeter involves complete elliptic integrals of the second kind, making Ramanujan's approximation invaluable for practical applications.