Area of an 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 1/2major axis.
b = is the semi-minor axis or1/2minor axis.



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

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