BYTETOOLS

Ellipse Calculator

Solve an ellipse from its semi-axes or its general equation: area, Ramanujan perimeter, eccentricity, foci, vertices, latus rectum and directrices.

47.1239
Area
25.527
Perimeter
0.8
Eccentricity
4
Focal distance c

Ellipse properties

x²/25 + y²/9 = 1

  • Center: (0, 0)
  • Orientation: major axis horizontal
  • Semi-major a = 5, semi-minor b = 3
  • Major axis length = 10, minor axis length = 6
  • Vertices: (-5, 0) and (5, 0)
  • Co-vertices: (0, -3) and (0, 3)
  • Foci: (-4, 0) and (4, 0)
  • Latus rectum 2b²/a = 3.6
  • Directrices: x = -6.25 and x = 6.25

Sketch

About the perimeter

An ellipse has no exact elementary perimeter formula, so this uses Ramanujan's second approximation P ≈ π(a + b)(1 + 3h / (10 + √(4 - 3h))) with h = (a - b)²/(a + b)². For a = 5, b = 3 it returns 25.527, and its relative error stays below about 10⁻⁹ for everyday axis ratios. Area πab, eccentricity √(1 - b²/a²) and the focal distance c = √(a² - b²) are all exact.

What is the Ellipse Calculator?

The ByteTools Ellipse Calculator takes either the semi-axes and center, or the general equation Ax² + Cy² + Dx + Ey + F = 0, and returns every property a conic-sections question asks for.

  • Two input modes: semi-axes with a center, or the general second-degree equation
  • Ramanujan's second approximation for perimeter, verified at a = 5, b = 3 giving 25.527
  • Exact area, eccentricity, focal distance, latus rectum and directrices
  • Detects the circle case and the hyperbola case and says which it is
  • Scaled sketch with both foci plotted
  • 100% private — every calculation runs in your browser

How to use the Ellipse Calculator

  1. 1

    Choose whether you are starting from semi-axes or from a general equation.

  2. 2

    Enter the semi-axis along x, the semi-axis along y and the center, or the five coefficients A, C, D, E and F.

  3. 3

    Read the area, perimeter, eccentricity and focal distance in the summary tiles.

  4. 4

    Open the properties panel for foci, vertices, co-vertices, latus rectum and directrices.

  5. 5

    Click Copy results to save the full solution.

About the Ellipse Calculator

The ByteTools Ellipse Calculator takes either the semi-axes and center, or the general equation Ax² + Cy² + Dx + Ey + F = 0, and returns every property a conic-sections question asks for. It works out which axis is major, then gives the area πab, the eccentricity √(1 - b²/a²), the focal distance c, the foci, vertices, co-vertices, latus rectum and directrices.

Perimeter uses Ramanujan's second approximation, because an ellipse has no exact elementary arc-length formula. For a = 5 and b = 3 it returns 25.527, matching the published value, and its relative error stays around 10⁻⁹ for ordinary axis ratios. The page says so openly rather than presenting the figure as exact.

Everything is computed locally in your browser using JavaScript, so no values are uploaded and the calculator keeps working offline. A scaled sketch shows the ellipse with both foci marked.

Frequently asked questions

What is the formula for the perimeter of an ellipse?

There is no exact formula using elementary functions — the true perimeter is an elliptic integral. This tool uses Ramanujan's second approximation, π(a + b)(1 + 3h/(10 + √(4 - 3h))) with h = (a - b)²/(a + b)². It is accurate to roughly nine decimal places for everyday ellipses.

How do you find the foci of an ellipse?

Compute c = √(a² - b²), where a is the semi-major axis and b the semi-minor axis, then move c units from the center along the major axis in both directions. For a = 5 and b = 3, c = 4, so an ellipse centered at the origin has foci at (-4, 0) and (4, 0).

What does the eccentricity of an ellipse tell you?

Eccentricity e = c/a measures how stretched the ellipse is. A value of 0 means a perfect circle, and values approaching 1 mean a long, thin ellipse. Earth's orbit has an eccentricity of about 0.0167, which is why it looks almost circular.

How do I know if an equation is an ellipse or a hyperbola?

Look at the signs of the x² and y² coefficients. Matching signs give an ellipse or circle; opposite signs give a hyperbola. If either coefficient is zero it is a parabola instead. This calculator checks the signs and redirects you when the equation is not an ellipse.

What is the latus rectum of an ellipse?

It is the chord through a focus that runs perpendicular to the major axis, with length 2b²/a. For a = 5 and b = 3 that is 3.6. It is a handy measure of how wide the curve is at the focus, and it appears often in orbital mechanics.

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