What is an ellipse?
An ellipse is a closed curve on a plane that looks like a stretched circle. It is formally defined as the set of all points for which the sum of distances to two fixed points (called foci) is constant. When both foci coincide at one point, the ellipse becomes a circle.
Ellipses appear everywhere in nature and engineering — from planetary orbits (Kepler's first law) to the cross-sections of cylinders and cones, to the shape of stadium tracks and satellite dishes.
Ellipse standard form
The equation of an ellipse centred at point (c₁, c₂) is:
(x − c₁)² / a² + (y − c₂)² / b² = 1
Where:
- a — semi-axis along the horizontal (x) direction
- b — semi-axis along the vertical (y) direction
- (c₁, c₂) — coordinates of the center
If a > b the ellipse is horizontal (wider than tall). If b > a it is vertical (taller than wide). When a = b the ellipse is a circle.
Ellipse area formula
The area of an ellipse is calculated using the semi-major axis a and semi-minor axis b:
A = π × a × b
This is a natural generalization of the circle area formula A = πr², recovered when a = b = r.
Ellipse perimeter (circumference)
Unlike the area, the exact perimeter of an ellipse has no simple closed form and requires an infinite series. Our calculator uses the highly accurate Ramanujan approximation:
P ≈ π(a + b) [1 + 3h / (10 + √(4 − 3h))] h = ((a − b) / (a + b))²
This approximation is accurate to within 0.00001% for most practical ellipses.
Eccentricity of an ellipse
Eccentricity e measures how "stretched" the ellipse is — how far it deviates from a perfect circle:
e = √(1 − (b/a)²) where a ≥ b (semi-major axis)
- e = 0: perfect circle
- 0 < e < 1: ellipse (closer to 1 → more elongated)
- e = 1: parabola (degenerate case)
Center, foci, and vertices of an ellipse
For a horizontal ellipse centred at (c₁, c₂) with a ≥ b:
- Center: (c₁, c₂)
- Foci: F₁ = (c₁ + c, c₂) and F₂ = (c₁ − c, c₂), where c = a × e
- Vertices: V₁ = (c₁ + a, c₂) and V₂ = (c₁ − a, c₂)
- Co-vertices: B₁ = (c₁, c₂ + b) and B₂ = (c₁, c₂ − b)
- Semi-latus rectum: l = b² / a
For a vertical ellipse (b > a) the foci lie on the y-axis instead: F₁ = (c₁, c₂ + c) and F₂ = (c₁, c₂ − c).
Units supported
This ellipse calculator supports both metric and US imperial measurement systems:
- Metric: mm (millimetres), cm (centimetres), m (metres), km (kilometres)
- US Imperial: in (inches), ft (feet), yd (yards), mi (miles)
Area results are shown in squared units (e.g. cm², ft²). All other measurements use the selected linear unit.
FAQs
What is the difference between the semi-major and semi-minor axis?
The semi-major axis is the longer of the two half-axes of an ellipse, and the semi-minor axis is the shorter one. Together they fully define the shape of the ellipse. Our calculator accepts a (horizontal half-axis) and b (vertical half-axis) and automatically identifies which is major and which is minor.
How do I find the foci of an ellipse?
The focal distance from the center is c = a × e = √(a² − b²) (where a is the semi-major axis). For a horizontal ellipse the foci are at (±c, 0) relative to the center; for a vertical ellipse they are at (0, ±c).
What is the semi-latus rectum?
The semi-latus rectum l = b²/a is the half-chord drawn perpendicular to the major axis through one focus. It describes the "width" of the ellipse at the focus point and appears in orbital mechanics (e.g. Kepler's equation for planetary orbits).
Can this calculator handle a circle?
Yes! When a = b the ellipse degenerates into a circle. The calculator will label the orientation as "Circle", display eccentricity = 0, and the area formula simplifies to A = πr².