# Area of an Ellipse

Ellipse

An ellipse is a curved line such that the sum of the distance of any point in it from two fixed points is constant. In the figure $F$ and $F’$ are the two fixed points. The fixed points are called foci. The point $O$ is the center of the ellipse. $NA$ is the major axis and $BM$ is the minor axis. $OA$ and $OB$ are the semi-axes.

Area of an Ellipse

The area of a circle is given by the formula $A = \pi {r^2}$, which may be written as $A = \pi r{\text{ }}r$. If a circle is flattened it will take the form of an ellipse and the semi-axes of such an ellipse (like $OA$ and $OB$ in the given figure) will be the lengthened and shortened radii. If $a$ stands for $OA$ and $b$ stands for $OB$, it can be proved that the area of the ellipse can be found by substituting $ab$ and $rr$ in the formula for the area of the circle, which then gives the following formula for the area of an ellipse:

$\boxed{A = \pi ab}$
Where $a =$ is the semi-major axis or $1/2$major axis.
$b =$ is the semi-minor axis or$1/2$minor axis.

Example:

Find the radius of a circle equal in area as that of an ellipse whose axes are $2$cm and $14$cm.

Solution:
Here $a = \frac{{30}}{2} = 15$,   $b = \frac{{26}}{2} = 13$
$\therefore$ area of the ellipse $A = \pi ab = 3.1416 \times 15 \times 13 = 612.6$