Ellipse area calculator

What is the area of an ellipse?

An ellipse is a geometric shape representing a closed plane curve, obtained by intersecting a cylindrical or conical surface with a plane. The shape of an ellipse resembles a flattened circle and occupies an essential place in geometry, physics, and astronomy. The area of an ellipse is a two-dimensional measure indicating how much space the ellipse occupies on a plane.

A specific formula calculates the area of an ellipse, considering its two main axes — the major and the minor. The major axis stretches through the ellipse’s longest length, while the minor axis runs through the shortest length. These axes help determine the size of the ellipse and, consequently, its area.

Basic characteristics of an ellipse

An ellipse is characterized by several essential parameters, with its main axes as two of them:

• Major axis (2a): This is the length of the longest line running through the center of the ellipse and joining its opposite sides.
• Minor axis (2b): This is the length of the shortest line perpendicular to the major axis and also passing through the center of the ellipse.

These lines represent the diameters of the ellipse, but since they differ, they should not be confused with the diameter of a circle. The ratio between the lengths of these axes defines the ellipticity of the shape.

Application of ellipse areas

The area of an ellipse is important in various scientific and engineering fields. For example:

1. Astronomy: Planets in our solar system describe elliptical orbits around the Sun. Knowing the area of these orbits assists scientists in calculating celestial bodies’ motion paths and determining orbital parameters.

2. Optics: Computing the sizes of elliptical lenses and mirrors requires knowledge of areas. This is important for creating certain types of telescopes and optical devices.

On our website, you will also find the Square footage calculator, which allows you to calculate the areas of various geometric shapes and use the results to compute the cost of materials needed for projects, considering the area.

Formula

The area $A$ of an ellipse is calculated using the following formula:

$A = \pi \times a \times b$

where:

• $\pi$ is a mathematical constant (approximately equal to 3.14159)
• $a$ is the length of the semi-major axis (half of the major axis)
• $b$ is the length of the semi-minor axis (half of the minor axis)

This formula is analogous to the circle area formula $\pi \times r^2$, except that the radii are replaced with the two semi-axes of the ellipse.

Calculation Examples

Let’s examine some calculations of the area of an ellipse with various semi-axis values.

1. Example 1: Suppose the major axis is 10 cm, and the minor axis is 6 cm. The semi-axes are:

   $a = \frac{10}{2} = 5$ cm

   $b = \frac{6}{2} = 3$ cm

   Ellipse area:

   $A = \pi \times 5 \times 3 \approx 47.12 \text{ cm}^2$

2. Example 2: For an ellipse with a major axis of 8 m and a minor axis of 4 m:

   $a = \frac{8}{2} = 4$ m

   $b = \frac{4}{2} = 2$ m

   Area:

   $A = \pi \times 4 \times 2 = 8\pi \approx 25.13 \text{ m}^2$

Notes

• An ellipse can be used to model many physical systems and processes. For example, some planets have elliptical orbits.
• If the minor and major axes approach each other, the ellipse can transform into a circle. In such cases, the standard area formula for a circle is used.
• This calculator allows converting measurement units and performing calculations using different length and area units, facilitating work with various measurement systems.

FAQs

How is the area of an ellipse calculated if only the perimeter or eccentricity is known?

If you only have peripheral parameters such as the perimeter or eccentricity, additional information is required to calculate the area. The area does not directly depend on these parameters without knowing the lengths of the axes.

How does an ellipse differ from an oval?

An ellipse is a mathematically defined shape with a formal equation basis. An oval, on the other hand, can have any closed form without a clear formula describing its shape.

How do you determine the major or minor axis if the ellipse is positioned at an angle?

One can use rotation of the coordinate system or image analysis to establish the orientation of the ellipse’s axes in space.

How do you calculate the area of an ellipse with axes of 12 cm and 8 cm?

The lengths of the ellipse’s semi-axes are:

$a = \frac{12}{2} = 6$ cm

$b = \frac{8}{2} = 4$ cm

Using the area formula for an ellipse:

$A = \pi \times 6 \times 4 \approx 75.40 \text{ cm}^2$

Thus, the area of the ellipse is approximately 75.40 square centimeters.

What measurement units can be used for area?

The area can be measured in square millimeters (mm²), square centimeters (cm²), square meters (m²), hectares, among others. Translating the area value from one measurement unit to another requires considering the factors for each system of measurement. For example, 1 m² = 10,000 cm².