Area of Ellipse

An ellipse is a two-dimensional shape that looks like an elongated circle. It has two axes: the major axis, which is the longest diameter, and the minor axis, which is the shortest diameter.

The area of the ellipse is the total space it encloses.

To find the area of the ellipse, we use the formula: π × a × b, where a and b are the semi-major and semi-minor axes, respectively.

Derivation of the formula:- An ellipse is essentially a stretched circle, and its area can be thought of as a scaled version of a circle's area. The area of a circle is πr². For an ellipse, the radius is replaced by the semi-major and semi-minor axes. Therefore, the area of the ellipse = π × a × b

We can find the area of the ellipse using the formula where the semi-major and semi-minor axes are commonly used. The area of the ellipse is calculated as follows:

Method Using the Semi-Major and Semi-Minor Axes

If the semi-major axis a and the semi-minor axis b are given, we find the area of the ellipse using the formula: Area = π × a × b

For example, if a and b are 5 cm and 3 cm, respectively, what will be the area of the ellipse? Area = π × a × b = π × 5 × 3 = 15π The area of the ellipse is 15π cm²

We measure the area of an ellipse in square units. The measurement depends on the system used:

In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²)

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²)

There are no special formulas for the area of an ellipse since it is calculated using the semi-major and semi-minor axes. However, understanding the orientation and the axes can be critical:

Case 1: Circular Shape If the ellipse becomes a circle (a = b), the area is calculated using the formula for a circle, Area = πr², where r is the radius.

Case 2: Rotated Ellipse If the ellipse is rotated but the axes are given, use the formula Area = π × a × b for the calculation.

To ensure that you get correct results while calculating the area of the ellipse, here are some tips and tricks you should know about:

• The semi-major and semi-minor axes are distinct and should not be confused.
   
• Ensure you use the correct units when calculating the area, and convert if necessary to maintain consistency. π is approximately 3.14159, but using a calculator with a π function is more precise.

• Area: The total space occupied by a shape. It tells us how much surface a shape covers.

• Ellipse: A geometrical shape that resembles a stretched circle, defined by two axes.

• Semi-Major Axis: The longest radius of an ellipse, half of the major axis.

• Semi-Minor Axis: The shortest radius of an ellipse, half of the minor axis.

• Pi (π): A mathematical constant approximately equal to 3.14159, used in calculations involving circles and ellipses.