Ellipse Calculator
Calculate ellipse area, perimeter and eccentricity. Solve for semi-major or semi-minor axis with Ramanujan perimeter approximation.
What is the Ellipse Calculator?
The Ellipse Calculator is a free online geometry tool for computing ellipse area, perimeter, and eccentricity. It uses the formula A = πab for area and Ramanujan's approximation for perimeter. You can also solve for the semi-major or semi-minor axis when area and the other axis are known.
This calculator is useful for students, engineers, architects, and designers working with elliptical shapes in astronomy, engineering, construction, and sports. For related three-dimensional shapes, use the Ellipsoid Calculator to compute volume and surface area.
How to Use the Ellipse Calculator
Select what you want to solve for from the dropdown menu. Enter the semi-axes (half of the full diameters) of the ellipse. The calculator instantly computes the result with a detailed step-by-step breakdown and shows the eccentricity.
Understanding Ellipse Formulas
Area: A = πab, where a is the semi-major axis and b is the semi-minor axis.
Perimeter (Ramanujan): P ≈ π(a + b)(1 + 3h² / (10 + √(4 - 3h²))) where h = (a - b)/(a + b).
Eccentricity: e = √(1 - b²/a²), ranging from 0 (circle) to approaching 1 (highly elongated).
A circle is a special case of an ellipse. Use the Circle Calculator for circular shapes, or the Ellipse Circumference Calculator for an alternative perimeter approximation.
Applications of Ellipse Geometry
- Astronomy: planetary orbits follow elliptical paths (Kepler's first law)
- Engineering: elliptical gears, cams, and machine components
- Architecture: oval rooms, stadium shapes, decorative arches
- Sports: running tracks and athletic fields
- Physics: elliptical machine paths and orbital mechanics
Frequently Asked Questions
How do you calculate the area of an ellipse?
Multiply π by the two semi-axes: A = πab. For an ellipse with a = 5 m and b = 3 m, A = 15π ≈ 47.12 m². The formula generalizes circle area πr².
What is the formula for the perimeter of an ellipse?
There is no exact closed-form formula. Ramanujan's approximation P ≈ π[3(a+b) - √((3a+b)(a+3b))] is accurate to better than 0.4% for any eccentricity. This calculator uses Ramanujan's second approximation for even better accuracy.
What is eccentricity of an ellipse?
e = √(1 - b²/a²), where a is the semi-major axis. Eccentricity 0 means a circle (a = b). As e approaches 1 the ellipse becomes very elongated. For a = 5 and b = 3, e = 0.8.
How is an ellipse different from a circle?
A circle is a special ellipse where both semi-axes are equal (a = b = r). Ellipses have two different axis lengths and two foci, while circles have one center point. Every circle is an ellipse, but only ellipses with a = b are circles.
How do you find the semi-major axis given area?
Rearrange A = πab to a = A / (πb). For A = 47.12 m² and b = 3 m, a = 47.12 / (3π) ≈ 5 m.
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