Ellipse Calculator

Ellipse Calculator

Ellipse Calculator is an online tool that helps you perform various calculations related to ellipses such as area and perimeter

What is Ellipse?

An ellipse is a closed curve in a two-dimensional plane resembling an elongated circle. It is defined by two foci which are two fixed points inside the ellipse. From any point on the ellipse, the sum of the distances to the two foci equals the major axis and remains constant.

Area of an Ellipse

The area A of an ellipse can be determined by using the formula given below:

A = π × r₁ × r₂

Where:

• A is the area of the ellipse
• r₁ is the length of the axis
• r₂ is the length of the other axis

The perimeter of an Ellipse

Finding the exact closed-form formula for the perimeter (P) of an ellipse is challenging, and it involves elliptic integrals. However, an approximate formula for the perimeter, known as Ramanujan's formula, provides a good estimation:

P ≈ [ 2 × π × Sqrt ((r₁² + r₂²)/2)]

Where:

• P is the perimeter of the ellipse
• r₁ is the length of the axis.
• r₂ is also the length of the other axis.

It is essential to note that Ramanujan's formula is an approximation and may not yield an exact value, especially for ellipses with extreme eccentricities. For more precise calculations, numerical methods or specialized software may be required.

The volume of an Ellipse

The volume of an ellipse can be determined by following the steps given below:

Volume = [ (4/3) ×π×r₁×r₂×r₃]

• r₁ is the length of the axis
• r₂ is the length of the axis
• r₃ is the length of the axis

Ellipse Characteristics

The shape of an ellipse is defined by its eccentricity which measures how elongated the ellipse is.

• e = 0 represents a perfect circle (the two foci coincide at the center).
• 0 < e < 1 indicates an ellipse with varying degrees of elongation.
• e = 1 represents a degenerate ellipse, where the two foci are at the ends of the major axis, and the ellipse becomes a line segment.

Applications of Ellipses

• Ellipses find applications in various fields, including astronomy, engineering, and art.
• Celestial orbits of planets, comets, and satellites are often elliptical.
• Antennas and reflectors in satellite dishes and telescopes have elliptical shapes for specific focusing properties.
• Understanding these characteristics allows us to use ellipses effectively in various practical applications and artistic representations.

How to find the area, perimeter, and volume of the ellipse?

Example 

Calculate the area, perimeter, and volume of an ellipse when r₁ = 21, r₂ = 54, r₃ = 18.

Solution

Area:

A = π × r₁ × r₂

A = 3.14 × 21 × 54

A = 3560.76

Perimeter:

P = [ 2 × π × Sqrt ((r₁² + r₂²)/2)]

P = [ 2 × 3.14 × Sqrt ((21² + 54²)/2)]

P = [ 2 × 3.14 × Sqrt ((3357)/2)]

P = 2 × 3.14 × Sqrt (1678.5)

P = 2 × 3.14 × 40.96

P = 257.2288

Volume:

V= [ (4/3) ×π×r₁×r₂×r₃]

V = [ (4/3) ×3.14×21×54×18]

V = (4/3) × 64093.68

V = 256374.72 / 3

V = 85458.24