Ellipse Calculator

What is an Ellipse?

An ellipse is a closed curved shape formed by all points in a plane for which the sum of the distances to two fixed points, called foci, remains constant. An ellipse looks similar to a stretched circle and is one of the most important shapes in geometry, engineering, astronomy, and physics.

The longest diameter of an ellipse is known as the major axis, while the shortest diameter is called the minor axis. The distances from the center to the ends of these axes are known as the semi-major radius and semi-minor radius respectively.

Ellipses are commonly found in planetary orbits, satellite paths, architectural designs, optics, and mechanical engineering applications.

Ellipse Formulas

Area of an Ellipse

The area enclosed by an ellipse is calculated using:

Area = π × a × b

where:

• a = semi-major radius
• b = semi-minor radius
• π ≈ 3.141592653589793

Perimeter of an Ellipse

An exact perimeter formula is complex, so the calculator uses the commonly accepted approximation:

Perimeter ≈ 2π√((a² + b²)/2)

Volume of an Ellipsoid

When an ellipse extends into three dimensions, it forms an ellipsoid. The volume is:

Volume = (4/3)πabc

where:

• a = first radius
• b = second radius
• c = third radius

Properties of an Ellipse

• Every circle is a special type of ellipse.
• An ellipse has two foci.
• The sum of distances from any point on the ellipse to the two foci remains constant.
• The major axis is always longer than or equal to the minor axis.
• The eccentricity of an ellipse lies between 0 and 1.

Real-Life Applications of Ellipses

• Planetary and satellite orbits.
• Optical reflectors and telescope mirrors.
• Architectural arches and domes.
• Mechanical gears and machine components.
• Computer graphics and CAD systems.
• Acoustic and lighting design.