Ellipse Area Calculator Formula

Understand the math behind the ellipse area calculator. Each variable explained with a worked example.

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Formulas Used

Area

area = pi * a_axis * b_axis

Circumference

circumference = pi * (3 * (a_axis + b_axis) - sqrt((3 * a_axis + b_axis) * (a_axis + 3 * b_axis)))

Eccentricity

eccentricity = a_axis >= b_axis ? sqrt(1 - pow(b_axis, 2) / pow(a_axis, 2)) : sqrt(1 - pow(a_axis, 2) / pow(b_axis, 2))

Variables

VariableDescriptionDefault
a_axisSemi-major Axis (a)6
b_axisSemi-minor Axis (b)4

How It Works

Ellipse Calculations

Area

Area = pi × a × b

Where a is the semi-major axis and b is the semi-minor axis.

Circumference (Ramanujan Approximation)

C ≈ pi × (3(a+b) - sqrt((3a+b)(a+3b)))

There is no exact closed-form formula for the circumference of an ellipse. Ramanujan's approximation is remarkably accurate.

Eccentricity

e = sqrt(1 - b²/a²) (when a ≥ b)

Eccentricity ranges from 0 (circle) to 1 (degenerate line).

Worked Example

Ellipse with semi-major axis 6 and semi-minor axis 4.

a_axis = 6b_axis = 4
  1. 01Area = pi × 6 × 4 ≈ 75.3982
  2. 02Circumference ≈ pi × (30 - √(22 × 16)) ≈ 31.7302
  3. 03Eccentricity = √(1 - 16/36) ≈ 0.7454

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