Ellipse Calculator

Calculate area, perimeter, eccentricity, foci, and directrix of an ellipse with step-by-step solutions.

Enter Ellipse Dimensions

Results

Area
--
square units
Perimeter (Ramanujan)--
Eccentricity--
Linear Eccentricity (c)--
Focus 1--
Focus 2--
Directrix 1--
Directrix 2--
Semi-latus Rectum--

Step-by-Step Solution

A = pi x a x b

What Is an Ellipse?

An ellipse is a closed curve in a plane that resembles a flattened circle. It is defined as the set of all points where the sum of the distances from two fixed points (called foci) is constant. Every ellipse has two axes of symmetry: the major axis (the longest diameter) and the minor axis (the shortest diameter). The semi-major axis (a) is half the major axis, and the semi-minor axis (b) is half the minor axis.

Ellipse Formulas

Area

The area enclosed by an ellipse with semi-axes a and b.

A = pi x a x b

Perimeter (Ramanujan)

An accurate approximation of the ellipse perimeter.

C = pi(3(a+b) - sqrt((3a+b)(a+3b)))

Eccentricity

Measures how much the ellipse deviates from a perfect circle (0 to 1).

e = sqrt(1 - b2/a2)

Foci

Two fixed interior points. The linear eccentricity c is the distance from center to each focus.

c = sqrt(a2 - b2)

Directrix

Two lines perpendicular to the major axis, located at distance a/e from the center.

x = +/- a/e = +/- a2/c

Semi-latus Rectum

The distance from a focus to the ellipse along a line perpendicular to the major axis.

l = b2/a

Understanding Eccentricity

The eccentricity of an ellipse is a number between 0 and 1 that describes how elongated the ellipse is. When e = 0, the ellipse is a perfect circle. As the eccentricity increases toward 1, the ellipse becomes more stretched and elongated. Earth's orbit around the Sun, for example, is an ellipse with eccentricity approximately 0.0167, which is very close to circular.

Relationship Between a, b, and c

For any ellipse, the semi-major axis (a), semi-minor axis (b), and the linear eccentricity (c) satisfy the fundamental relationship: a2 = b2 + c2. This is analogous to the Pythagorean theorem and connects all three key measurements of the ellipse.

Applications of Ellipses

  • Astronomy: Planets, comets, and satellites follow elliptical orbits as described by Kepler's first law of planetary motion.
  • Engineering: Elliptical gears are used for variable-speed mechanisms. Elliptical cross-sections appear in aerodynamic designs.
  • Architecture: Elliptical arches and domes are used in buildings for both aesthetics and structural efficiency.
  • Optics: Elliptical mirrors and lenses focus light from one focus point to the other, used in telescopes and medical equipment.
  • Acoustics: Whispering galleries exploit the reflective property of ellipses, where sound from one focus is reflected to the other.

Tips for Using This Calculator

  • The semi-major axis (a) must always be greater than or equal to the semi-minor axis (b). If you enter a smaller value for a, the calculator will swap them automatically.
  • For a circle, set a = b. The eccentricity will be 0 and the foci will coincide at the center.
  • The perimeter uses Ramanujan's first approximation, which is accurate to within 0.02% for most ellipses.
  • All results assume the ellipse is centered at the origin (0, 0).