What is the circumference of an ellipse?

The circumference (perimeter) of an ellipse is the total length of its boundary. Unlike a circle, there is no simple exact formula for an ellipse's circumference. It requires either infinite series calculations or approximation formulas like those developed by mathematician Ramanujan.

Why is there no exact formula for ellipse circumference?

The ellipse circumference involves an elliptic integral, which cannot be expressed using elementary functions. This is why mathematicians have developed various approximation formulas. The Ramanujan II approximation provides accuracy better than 0.0001% for most practical applications.

What is the most accurate formula for ellipse circumference?

The Ramanujan II approximation formula C = pi(a+b)(1 + 3h/(10 + sqrt(4-3h))) where h = ((a-b)/(a+b))^2 provides excellent accuracy with errors less than 0.0001% for most ellipses. For even higher precision, an infinite series expansion using elliptic integrals can be computed.

What is the eccentricity of an ellipse?

Eccentricity (e) measures how elongated an ellipse is. It ranges from 0 (a perfect circle) to almost 1 (a very elongated ellipse). It is calculated as e = sqrt(1 - (b/a)^2), where a is the semi-major axis and b is the semi-minor axis. Higher eccentricity means a more stretched ellipse.

What are the semi-major and semi-minor axes of an ellipse?

The semi-major axis (a) is half the longest diameter of the ellipse, while the semi-minor axis (b) is half the shortest diameter. Together they define the size and shape of the ellipse. By convention, a is always greater than or equal to b.

Ellipse Circumference Calculator - Calculate Ellipse Perimeter with Multiple Formulas

Semi-major axis (a):

Half of the longest diameter

Semi-minor axis (b):

Half of the shortest diameter