Ellipse Area Calculator Formula
Understand the math behind the ellipse area calculator. Each variable explained with a worked example.
Formulas Used
Area
area = pi * a_axis * b_axisCircumference
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
| Variable | Description | Default |
|---|---|---|
a_axis | Semi-major Axis (a) | 6 |
b_axis | Semi-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.
- 01Area = pi × 6 × 4 ≈ 75.3982
- 02Circumference ≈ pi × (30 - √(22 × 16)) ≈ 31.7302
- 03Eccentricity = √(1 - 16/36) ≈ 0.7454
Frequently Asked Questions
What is an ellipse?
An ellipse is a stretched circle. It is the set of all points where the sum of distances to two fixed points (foci) is constant.
What is eccentricity?
Eccentricity measures how "stretched" the ellipse is. e = 0 is a perfect circle, and as e approaches 1, the ellipse becomes more elongated.
Why is the circumference formula approximate?
Unlike circles, there is no exact algebraic formula for the perimeter of an ellipse. It requires elliptic integrals. Ramanujan's formula is an excellent approximation.
Learn More
Guide
How to Calculate Area of Shapes - Complete Guide
Learn how to calculate the area of common shapes including rectangles, triangles, circles, trapezoids, parallelograms, and ellipses with formulas and examples.
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Open Ellipse Area Calculator