Ellipse Calculator

Calculate area, perimeter, eccentricity, focal distance, and other properties of an ellipse. Perfect for geometry, orbital mechanics, and engineering applications.

Half of the longest diameter

Half of the shortest diameter

Area = PIab Perimeter ≈ PI(a+b)[1 + 3h/(10+sqrt(4-3h))] where h = (a-b)^2/(a+b)^2 Eccentricity = sqrt(1 - b^2/a^2) Focal Distance = sqrt(a^2 - b^2) Standard Form: x^2/a^2 + y^2/b^2 = 1 Where: a = semi-major axis (half longest diameter) b = semi-minor axis (half shortest diameter) e = eccentricity (0 to 1) c = distance from center to focus
Example 1 (Moderately Eccentric Ellipse): Semi-major axis (a) = 5 cm Semi-minor axis (b) = 3 cm Area = PI * 5 * 3 = 47.12 cm^2 Perimeter ≈ 25.53 cm (Ramanujan) Eccentricity = sqrt(1 - 9/25) = 0.8 Focal distance = sqrt(25 - 9) = 4 cm Foci are 4 cm from center on major axis Example 2 (Nearly Circular): a = 5 cm, b = 4.9 cm Area = 76.97 cm^2 Perimeter ≈ 31.26 cm Eccentricity = 0.14 (nearly circular) Focal distance = 0.99 cm Example 3 (Planetary Orbit): Earth's orbit: a = 149.6 million km, b = 149.58 million km Eccentricity ≈ 0.0167 (very low, nearly circular)

What are the formulas for an ellipse?

Area = PIab (where a and b are semi-major and semi-minor axes). Perimeter ≈ PI[3(a+b) - sqrt((3a+b)(a+3b))] (Ramanujan approximation). Eccentricity = sqrt(1 - b^2/a^2). The ellipse has two focal points at distance c = sqrt(a^2 - b^2) from center.

What is the difference between major and minor axes?

The major axis is the longest diameter of the ellipse (length = 2a), while the minor axis is the shortest diameter (length = 2b). The semi-major axis (a) is half the major axis, and semi-minor axis (b) is half the minor axis. By convention, a >= b.

How do you calculate ellipse eccentricity?

Eccentricity (e) measures how "stretched" an ellipse is: e = sqrt(1 - b^2/a^2), where a is semi-major axis and b is semi-minor axis. e = 0 means a perfect circle, while e approaching 1 means very elongated. Example: a=5, b=3 gives e = sqrt(1 - 9/25) = 0.8.

What is the focal distance of an ellipse?

The focal distance (c) is the distance from the center to each focus point: c = sqrt(a^2 - b^2). The two foci are located on the major axis. The sum of distances from any point on the ellipse to both foci equals 2a (constant). Example: a=5, b=3 gives c = 4.

How accurate is the ellipse perimeter formula?

The exact perimeter requires elliptic integrals (no simple formula). Ramanujan's approximation PI[3(a+b) - sqrt((3a+b)(a+3b))] is accurate to within 0.01% for most ellipses. Other approximations include PI(a+b) (rough) and 2PIsqrt((a^2+b^2)/2) (moderate accuracy).

What are real-world examples of ellipses?

Planetary orbits (Earth orbits Sun in an ellipse), satellite orbits, elliptical gears, racetrack shapes, whispering galleries, lithotripsy (kidney stone treatment), elliptical trainers, amphitheaters, architectural arches, and pools. Many natural and engineered systems follow elliptical patterns.

How do you draw an ellipse?

String method: Place two pins at the foci, loop a string around both, keep string taut with a pencil and trace. The string length equals 2a (major axis). Mathematical method: Use equation (x^2/a^2) + (y^2/b^2) = 1. Digital: Use graphics software ellipse tool with specified axes.

What happens when a = b in an ellipse?

When semi-major axis equals semi-minor axis (a = b), the ellipse becomes a circle with radius r = a = b. Eccentricity becomes 0, focal distance becomes 0 (foci merge at center), area = PIr^2, and perimeter = 2PIr. A circle is a special case of an ellipse.