Why is the circumference only approximate?

No elementary formula gives the exact perimeter of an ellipse — the true value is an elliptic integral. The 2π√((a² + b²)/2) form stays within a fraction of a percent as long as the ellipse is not extremely flat.

How do I convert a diameter into a semi-axis?

Halve it. A 10 × 6 ellipse has semi-axes 5 and 3, so its area is π × 15 ≈ 47.1239.

What is eccentricity used for?

It measures how stretched the ellipse is: e = √(1 − (b/a)²). A circle scores 0, Earth's orbit about 0.0167, and some comets exceed 0.9.

🔵 Ellipse Area Calculator

Calculate the area of an ellipse from its semi-major and semi-minor axes.

Semi-major axis - Long radius (a) Semi-minor axis - Short radius (b)

What is an Ellipse?

An ellipse is the set of all points in a plane where the sum of distances from two fixed points F₁ and F₂ (foci) is constant. For any point P: |PF₁| + |PF₂| = 2a (major axis).

Area Calculation

S = π × a × b

S: Area | a: semi-major axis (long radius) | b: semi-minor axis (short radius)

Circumference Calculation

L ≈ 2π × √((a² + b²) / 2)

Lambert approximation — approximation of the exact integral value

The circumference of an ellipse has no simple formula like a circle. The formula above is the most commonly used approximation.

Focal Distance

c = √(a² − b²)

Distance between foci F₁ and F₂ = 2c

Eccentricity

e = c / a = √(1 − (b/a)²)

e = 0 is a circle, 0 < e < 1 is an ellipse. The closer e is to 1, the more elongated the ellipse.

Equation of an Ellipse

x² / a² + y² / b² = 1

Standard form ellipse centered at the origin. If a > b > 0, the major axis lies on the x-axis.

Worked examples

An ellipse with semi-axes 5 and 3 has an area of π × 5 × 3 ≈ 47.1239 in² and a circumference of about 25.9081 in.

When both semi-axes are 4 the ellipse is a circle: area π × 16 ≈ 50.2655 and circumference 2π × 4 ≈ 25.1327 exactly.

What the result assumes

The area is exact, but the circumference shown is the approximation 2π√((a² + b²)/2), which stays slightly low for very stretched ellipses; Ramanujan's formula is closer. Enter semi-axes — half of each diameter — because the full lengths would quadruple the area. Values keep whatever unit you type, so linear inputs give square units out.

Frequently asked questions

Why is the circumference only approximate?: No elementary formula gives the exact perimeter of an ellipse — the true value is an elliptic integral. The 2π√((a² + b²)/2) form stays within a fraction of a percent as long as the ellipse is not extremely flat.
How do I convert a diameter into a semi-axis?: Halve it. A 10 × 6 ellipse has semi-axes 5 and 3, so its area is π × 15 ≈ 47.1239.
What is eccentricity used for?: It measures how stretched the ellipse is: e = √(1 − (b/a)²). A circle scores 0, Earth's orbit about 0.0167, and some comets exceed 0.9.