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 $$1/2$$major axis.
$$b = $$ is the semi-minor axis or$$1/2$$minor axis.



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

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