Ellipse Standard Form Calculator
Enter the semi-major axis (a), semi-minor axis (b), and center coordinates (h, k) of an ellipse to instantly get its standard-form equation, area, perimeter, eccentricity, foci, vertices, co-vertices, latus rectum (focal width), and directrices. A "show your work" panel walks through each formula step by step.
What is the standard form of an ellipse?
An ellipse in standard form is written as (x - h)² / a² + (y - k)² / b² = 1, where (h, k) is the center, a is the semi-major axis (longer half-axis), and b is the semi-minor axis (shorter half-axis). When the center is at the origin the formula simplifies to x² / a² + y² / b² = 1. This form makes it easy to read off the key geometric properties directly: center, vertices, and the size of each axis.
How to identify orientation: horizontal vs. vertical ellipse
If a > b, the major axis lies along the x-axis and the ellipse is wider than it is tall (horizontal). If b > a, the major axis lies along the y-axis and the ellipse is taller than it is wide (vertical). When a equals b the shape is a circle. The foci always lie on the major axis, at a distance c = sqrt(a² - b²) from the center for a horizontal ellipse, or c = sqrt(b² - a²) for a vertical one.
Foci, eccentricity, and the focal width
The eccentricity e = c / (semi-major axis) is a number between 0 and 1 that describes how stretched the ellipse is. At e = 0 you have a circle; as e approaches 1 the ellipse becomes increasingly flat. The foci are the two special interior points such that any point on the ellipse has the same total distance to both foci, equal to 2a (or 2b if vertical). The latus rectum (focal width) is the chord through either focus perpendicular to the major axis, with length 2b² / a for a horizontal ellipse.
Perimeter and area formulas
The area of any ellipse is exactly A = pi * a * b. The perimeter has no exact closed form in elementary functions, but Ramanujan's second approximation gives excellent results: P = pi(a + b)(1 + 3h / (10 + sqrt(4 - 3h))), where h = (a - b)² / (a + b)². This approximation is accurate to better than 1 part per 10,000 for all ellipses. For a circle (a = b), both formulas reduce to the familiar pi * r² and 2 * pi * r.
Directrices and their relationship to the foci
Every ellipse has two directrix lines, one on each side of the center, perpendicular to the major axis. For a horizontal ellipse they are vertical lines at x = h plus or minus a / e. The ratio of any point's distance to a focus divided by its distance to the corresponding directrix equals e, the eccentricity. This focus-directrix property is one of the classical conic-section definitions and links the ellipse to the parabola (e = 1) and the hyperbola (e > 1).
Eccentricity and ellipse shape
| Eccentricity e | Shape description | Note |
|---|---|---|
| e = 0 | Perfect circle | a = b |
| 0 < e < 0.3 | Nearly circular | Foci close to center |
| 0.3 - 0.7 | Moderate ellipse | Clearly non-circular |
| 0.7 - 0.9 | Elongated ellipse | Foci well off center |
| e approaching 1 | Very flat / degenerate | Foci near vertices |
How eccentricity (e) describes the shape of the ellipse.
Frequently asked questions
What is the difference between a and b in the standard form of an ellipse?
In the equation (x - h)² / a² + (y - k)² / b² = 1, a is the semi-axis along the x-direction and b is the semi-axis along the y-direction. The larger of the two is called the semi-major axis (giving the longer radius) and the smaller is the semi-minor axis. If a > b the ellipse is horizontal (wider), and if b > a it is vertical (taller).
How do I find the foci of an ellipse from standard form?
First compute c = sqrt(|a² - b²|). If a > b (horizontal ellipse), the foci are at (h - c, k) and (h + c, k). If b > a (vertical ellipse), they are at (h, k - c) and (h, k + c). If a = b the shape is a circle and the two foci coincide at the center.
What does eccentricity tell me about an ellipse?
Eccentricity (e) ranges from 0 to just below 1 for an ellipse. A value of 0 means a perfect circle; values near 1 describe very elongated, nearly flat ellipses. Eccentricity is defined as e = c / a, where c is the distance from the center to a focus and a is the semi-major axis. It tells you how far the ellipse deviates from circular.
Can I use this calculator for a vertical ellipse?
Yes. Enter the semi-axis values as they appear in the equation: if your equation has a larger denominator under the y term, enter that square root as b. The calculator detects orientation automatically: when b > a it treats the major axis as vertical, computes foci along y, and labels the output accordingly.
What is the latus rectum (focal width) of an ellipse?
The latus rectum is the chord drawn through a focus perpendicular to the major axis. Its length is 2b² / a for a horizontal ellipse (or 2a² / b for a vertical one). It is useful in optics and orbital mechanics because it defines the width of the conic at the focus, which governs properties like reflector dish shape and planetary orbit size at periapsis.
How do I convert an ellipse from general form to standard form?
Starting from Ax² + Cy² + Dx + Ey + F = 0, group the x and y terms separately, factor out the coefficients A and C, then complete the square for each group. Add the same values to both sides to keep the equation balanced. Finally divide through so the right-hand side equals 1. The result is (x - h)² / a² + (y - k)² / b² = 1.