Question

what is the area of eclipse

Answer 1

These semi-major axes are half the lengths of, respectively, the largest and smallest diameters of the ellipse---


For example, the following is a standard equation for such an ellipse centered at the origin:

(x2/A2) + (y2/B2) = 1.

The area of such an ellipse is

Area = Pi * A * B ,
a very natural generalization of the formula for a circle!

Presentation Suggestions:
If students guess this fact, ask them what they think the volume of an ellipsoid is!

The Math Behind the Fact:
One way to see why the formula is true is to realize that the above ellipse is just a unit circle that has been stretched by a factor A in the x-direction, and a factor B in the y-direction. Hence the area of the ellipse is just A*B times the area of the unit circle.

The formula can also be proved using a trigonometric substitution. For a more interesting proof, use line integrals and Green's Theorem in multivariable calculus.

Each of the above proofs will generalize to show that the volume of an ellipsoid with semi-axes A, B, and C is just

Answer 2

 area of an ellipse can be found by the following formula area = Πab 

where b is the distance from the center to a co vertex a is the distance from the center to vertex