Ellipse
The locus with a constant sum of distances to two foci — the true shape of planetary orbits.
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Formulas and key values
- Cartesian equation
- Parametric equations
- AreaIn the graph ≈ 5.027
- Perimeter · Ramanujan’s approximationIn the graph ≈ 8.277
- Foci
- Eccentricity
a and b are the semi-axes (a ≥ b) and c is the center-to-focus distance.
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Start the video01 · Definition
What is the Ellipse?
The ellipse is the set of points whose distances to two foci add to a constant 2a — the geometry behind the pins-and-string drawing method. Squash a circle by a uniform factor along one axis and you get an ellipse; that is exactly what the parametrization x = a·cos t, y = b·sin t says.
Kepler’s first law of 1609 made the ellipse the most important curve in the sky: every planet orbits the Sun along an ellipse with the Sun at one focus. Its reflective property — a signal leaving one focus arrives at the other — powers whispering galleries and the shock-wave lithotripters used to break kidney stones.
02 · Properties
Key properties
- Foci at (±c, 0) with c² = a² − b²; eccentricity e = c/a < 1.
- Area is exactly πab; the perimeter has no elementary closed form (Ramanujan’s approximation is the standard workaround).
- Sum of focal distances is constant: r₁ + r₂ = 2a.
- Reflective property: one focus maps to the other.
- A circle is the special case a = b (e = 0).
03 · Questions
Ellipse: common questions
- What is the equation of the Ellipse?
- Cartesian equation: (x^2)/(a^2) + (y^2)/(b^2) = 1; Parametric equations: x = a cos t, y = b sin t. a and b are the semi-axes (a ≥ b) and c is the center-to-focus distance.
- What is the perimeter of the Ellipse?
- Perimeter (Ramanujan’s approximation): L ≈ π[3(a + b) - √((3a + b)(a + 3b))]. a and b are the semi-axes (a ≥ b) and c is the center-to-focus distance.
- What is the area of the Ellipse?
- Area: A = π ab. a and b are the semi-axes (a ≥ b) and c is the center-to-focus distance.
04 · Labs
Interactive labs
Conic sections and eccentricity
How can one number turn a circle into an ellipse, a parabola and a hyperbola?
Focus and directrix · Eccentricity · Ellipse, parabola, hyperbola
Open labThe ellipse: a constant sum of distances to two foci
Why does a loop of string around two pins draw an ellipse, and where does b² = a² − c² come from?
Definition by two foci · Standard equation · b² = a² − c²
Open lab05 · Keep exploring