Ellipse Area and Perimeter Calculator
Enter semi-major and semi-minor axes. The Ellipse Area and Perimeter Calculator instantly computes exact area and highly accurate perimeter using Ramanujan's approximation.
About the Ellipse Area and Perimeter Calculator
The Ellipse Area and Perimeter Calculator is a precise, scientifically accurate tool that computes the area and perimeter of an ellipse using established mathematical formulas. This Ellipse Area and Perimeter Calculator provides the exact area πab and a highly accurate approximation for the perimeter (circumference) using Ramanujan's second approximation, which is accurate to better than 0.001% for all eccentricities. Proudly supported by Agri Care Hub.
Scientific Formulas
Perimeter (Ramanujan II): P ≈ π (a + b) [1 + 3h / (10 + √(4 - 3h))]
where h = (a - b)² / (a + b)²
Eccentricity e = √(1 - (b/a)²)
Why This Calculator Is Essential
Ellipses appear in many fields: • Planetary orbits (Kepler's laws) • Engineering (elliptical gears, pipes) • Architecture and design • Optics (elliptical mirrors) • Medical imaging (cross-sections) • Agriculture (field shapes, irrigation) The area is exact, while the perimeter has no closed-form expression — Ramanujan's approximation is the gold standard for high accuracy.
How to Use
- Enter semi-major axis a (longer radius).
- Enter semi-minor axis b (shorter radius, b ≤ a).
- Click “Calculate Ellipse Area and Perimeter”.
- Get area, perimeter, eccentricity, and more.
When Should You Use This Tool?
- Astronomy and orbital mechanics
- Engineering design and CAD
- Architecture and landscaping
- Physics and optics
- Agricultural field planning
- Teaching conic sections
Scientific Foundation
The area of an ellipse πab is exact and derived from affine transformation of a circle. The perimeter has no elementary closed form, but Ramanujan's approximations (1914) are remarkably accurate. The second approximation used here has relative error less than 10⁻⁵. Full theory at Ellipse Area and Perimeter on Wikipedia and mathematical handbooks.
Conclusion
The Ellipse Area and Perimeter Calculator brings professional conic section mathematics to your browser — with perfect accuracy and beautiful design. Whether you’re a student learning geometry, an engineer designing elliptical components, or an astronomer calculating orbits, this tool delivers precise results every time. For more math tools, visit Agri Care Hub.