Foci of Ellipse Calculator

Find the foci, eccentricity, directrices, and other properties of an ellipse from its semi-major and semi-minor axes.

Enter Ellipse Parameters

Result

Foci Coordinates
(-4, 0) and (4, 0)
c = 4
Semi-major axis (a) 5
Semi-minor axis (b) 3
Focal distance (c) 4
Focus 1 (-4, 0)
Focus 2 (4, 0)
Eccentricity (e) 0.8
Directrix 1 x = -6.25
Directrix 2 x = 6.25
Area 47.12389
Perimeter (approx.) 25.5268
Linear eccentricity (c/a) 0.8
Semi-latus rectum (l) 1.8

Step-by-Step Solution

c = sqrt(a^2 - b^2) = sqrt(25 - 9) = 4

Understanding the Foci of an Ellipse

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. The foci (singular: focus) are two special points located on the major axis of the ellipse, symmetrically placed about the center. The distance from the center to each focus is denoted c, and satisfies the fundamental relationship c2 = a2 - b2, where a is the semi-major axis and b is the semi-minor axis.

The standard form of an ellipse centered at the origin with a horizontal major axis is x2/a2 + y2/b2 = 1, where a > b > 0. For a vertical major axis, the roles of x and y are swapped.

Key Ellipse Formulas

Focal Distance

The distance from the center to each focus, derived from the Pythagorean relationship.

c = sqrt(a^2 - b^2)

Eccentricity

Measures how elongated the ellipse is. Ranges from 0 (circle) to 1 (degenerate).

e = c / a = sqrt(1 - b^2/a^2)

Directrices

Lines perpendicular to the major axis at distance a/e from the center.

x = +/- a/e = +/- a^2/c

Area of Ellipse

The area enclosed by the ellipse, a generalization of the circle area formula.

A = pi * a * b

Perimeter (Ramanujan)

An excellent approximation for the perimeter of an ellipse.

P ~ pi(3(a+b) - sqrt((3a+b)(a+3b)))

Semi-Latus Rectum

Half the length of the chord through a focus perpendicular to the major axis.

l = b^2 / a

The Reflective Property of Ellipses

One of the most remarkable properties of an ellipse is its reflective property: any ray emanating from one focus will reflect off the ellipse and pass through the other focus. This principle is used in whispering galleries (like the one in St. Paul's Cathedral), where sound from one focus travels along the elliptical ceiling and converges at the other focus. It is also the basis for lithotripsy, a medical procedure that uses focused sound waves to break kidney stones.

Ellipse vs Circle

A circle is a special case of an ellipse where a = b (both semi-axes are equal). In this case, c = 0, meaning both foci coincide at the center, and the eccentricity is 0. As b approaches 0 while a remains fixed, the ellipse becomes more elongated and the eccentricity approaches 1.

Planetary Orbits

Kepler's first law of planetary motion states that planets orbit the Sun in elliptical paths, with the Sun at one focus. The eccentricity of Earth's orbit is approximately 0.0167, making it nearly circular. In contrast, Pluto's orbit has an eccentricity of about 0.25, making it noticeably elliptical. Understanding the foci of ellipses is therefore essential in celestial mechanics and orbital calculations.

How to Find the Foci

  1. Identify the semi-major axis a (the larger value) and semi-minor axis b (the smaller value).
  2. Compute c = √(a2 - b2).
  3. For a horizontal ellipse centered at (h, k), the foci are at (h - c, k) and (h + c, k).
  4. For a vertical ellipse centered at (h, k), the foci are at (h, k - c) and (h, k + c).