Ellipse calculator
This ellipse calculator works from the semi-major and semi-minor axis – or from the full axes. With a = 5 and b = 3 the area is 47.12, the circumference 25.53 and the eccentricity 0.8, with the calculation steps.
Example: Known values: Semi-axes a and b · Semi-major axis a 5 · Semi-minor axis b 3 → Result: Area A = 47.12389 · Circumference C = 25.527. Source: Wikipedia – Ellipse (Fläche, Umfang). Updated: .
How it is calculated
An ellipse is the set of all points whose distances to two foci add up to a constant. It is described by the semi-major axis a and the semi-minor axis b. For a = b it is a circle.
The formulas
- Area: A = π · a · b
- Circumference (Ramanujan approximation): C ≈ π · (a + b) · (1 + 3h ÷ (10 + √(4 − 3h))) with h = ((a − b) ÷ (a + b))²
- Focal distance: c = √(a² − b²)
- Eccentricity: ε = c ÷ a
- Semi-latus rectum: p = b² ÷ a
How to use it
Enter the semi-axes (or the full axes). Example: a = 5 and b = 3 give A = π · 15 = 47.12, c = 4, ε = 0.8 and a circumference of about 25.53. Unlike the area, the circumference of an ellipse has no simple closed formula; Ramanujan’s approximation is accurate to many digits. The unit is up to you; the area is in the square of the chosen unit.
Frequently asked questions
How do you calculate the area of an ellipse?
A = π · a · b with the two semi-axes. For a = 5 and b = 3 that is 47.12.
How do you calculate the circumference of an ellipse?
An exact formula needs elliptic integrals. The tool uses Ramanujan’s accurate approximation; for a = 5 and b = 3, C ≈ 25.53.
What is the difference between a semi-axis and an axis?
The axis runs straight across the ellipse, the semi-axis from the centre to the edge. 2a is the major axis, 2b the minor axis.
What does eccentricity mean?
It measures how far the ellipse departs from a circle: ε = c ÷ a with c = √(a² − b²). A circle has ε = 0; the flatter the ellipse, the closer to 1.
Where are the foci?
On the major axis at distance c = √(a² − b²) from the centre. For a = 5 and b = 3 they lie 4 units left and right of the centre.
Sources and legal basis
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