Ellipse Standard Form Calculator

Enter an ellipse's semi-axis lengths and center coordinates to get its standard-form equation (x-h)²/a² + (y-k)²/b² = 1, along with eccentricity, foci, and area.

Quick Facts

Standard form
(x−h)²/a² + (y−k)²/b² = 1
Center (h,k); a is the semi-axis along x, b the semi-axis along y.
Eccentricity
e = c/A, c = √(A²−B²)
A is the larger semi-axis, B the smaller; e=0 is a circle, e→1 is very flat.
Area formula
A = πab
Exact — no approximation needed, unlike perimeter.
Perimeter (approx.)
Ramanujan: π(a+b)[1+3h/(10+√(4−3h))]
No elementary exact formula exists; h=((a−b)/(a+b))².

Your Results

Calculated
Standard Form Equation
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(x−h)²/a² + (y−k)²/b² = 1
Eccentricity
-
e = c/A, where c = √(A²−B²)
Foci
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Located on the major axis, distance c from center
Area
-
A = πab

Ready

Enter the semi-axis lengths and center coordinates, then press Calculate.

Formula and Method for the Standard Form of an Ellipse

An ellipse is the set of all points where the sum of the distances to two fixed points, the foci, is constant. When centered at (h,k) with its axes parallel to the x- and y-axes, that shape is written in standard form as (x−h)²/a² + (y−k)²/b² = 1, where a is the semi-axis length in the x-direction and b is the semi-axis length in the y-direction. This calculator takes a, b, h, and k and derives the equation plus the ellipse's key properties: eccentricity, foci, vertices, and area.

How the calculation works

First the calculator compares a and b to find the semi-major axis A = max(a,b) and semi-minor axis B = min(a,b) — whichever of a or b is larger determines whether the ellipse is wider (major axis horizontal) or taller (major axis vertical). The distance from the center to each focus is c = √(A²−B²), derived from the same Pythagorean relationship that defines an ellipse's shape. Eccentricity is then e = c/A, a unitless number between 0 (a circle, where a = b and c = 0) and just under 1 (a very flattened ellipse). The two foci and two vertices sit on the major axis at distance c and A from the center; the two co-vertices sit on the minor axis at distance B. Area has an exact formula, A = πab, since an ellipse is a circle stretched by different factors along each axis. Perimeter has no elementary closed-form formula, so the calculator uses Ramanujan's well-known approximation, accurate to a tiny fraction of a percent for any a and b.

Common mistakes

  • Assuming a is always the major axis: in x²/a² + y²/b² = 1, the major axis is horizontal only if a > b. If b > a, the ellipse is taller than it is wide and the major axis runs vertically.
  • Sign errors with the center: the standard form uses (x−h) and (y−k), so a center at h = −3 makes the term (x−(−3)) = (x+3), not (x−3).
  • Treating perimeter like area: unlike A = πab, there is no simple exact formula for an ellipse's perimeter — using 2π√((a²+b²)/2) or similar shortcuts can introduce noticeable error for elongated ellipses.

Real-world applications

  • Astronomy and orbital mechanics: planetary and satellite orbits are ellipses with the sun or planet at one focus (Kepler's first law), and eccentricity describes how circular or stretched an orbit is.
  • Architecture and acoustics: elliptical rooms and "whispering galleries" reflect sound from one focus to the other, a direct consequence of the focal-point definition of an ellipse.
  • Optics: elliptical mirrors and reflectors focus light or radiation from one focal point to the other.
  • Engineering and design: cams, gears, and structural arches use elliptical curves where a smooth, non-circular profile is required.

Frequently Asked Questions

What is the standard form equation of an ellipse?
The standard form is (x−h)²/a² + (y−k)²/b² = 1, where (h,k) is the center, a is the semi-axis length in the x-direction, and b is the semi-axis length in the y-direction. If a > b the major axis is horizontal; if b > a the major axis is vertical; if a = b the ellipse is a circle.
How do I find the foci of an ellipse from its standard form?
First find c using c = √(A²−B²), where A is the larger of a and b (the semi-major axis) and B is the smaller (the semi-minor axis). The two foci sit on the major axis at a distance c from the center: (h±c, k) for a horizontal major axis, or (h, k±c) for a vertical one.
What does eccentricity tell you about an ellipse?
Eccentricity e = c/A (with A the semi-major axis) measures how stretched an ellipse is. It ranges from e = 0, which is a perfect circle, up toward e = 1, which is an increasingly flattened ellipse. Planetary orbits, for example, typically have eccentricities close to 0.
How do you find the area and perimeter of an ellipse?
Area has an exact formula, A = πab. Perimeter has no elementary closed-form expression; this calculator uses Ramanujan's approximation, P ≈ π(a+b)[1 + 3h/(10+√(4−3h))] with h = ((a−b)/(a+b))², which is accurate to within a tiny fraction of a percent.

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