Area of an Ellipse

a

5

b

3

n

2

unit circle area = 3.14159

inner area = 1.

outer area = 4.

ellipse area = 47.1239

inner area = 15.

outer area = 60.

What is the area of an ellipse with equation

2

x

2

a

+

2

y

2

b

=1

?

This ellipse is the image of the unit circle

2

u

+

2

v

=1

under the transformation

x=au

,

y=bv

.

This transformation maps a square into a rectangle whose area is

ab

times the area of the square.

Unions of disjoint squares inside (or crossing) the circle are mapped into unions of disjoint rectangles inside (or crossing) the ellipse.

The area of the ellipse must therefore be

ab

times the area of the unit circle, or

πab

.

The Demonstration maps squares with side

1

n

into rectangles with sides

a

n

and

b

n

and computes inner and outer approximations to the areas of the unit circle and the image ellipse.

The difference between the outer and inner areas for the circle is less than

π

2

(h+1)

-

2

(h-1)

=4πh

, where

h=

2

n

, and hence the difference between the outer and inner areas for the ellipse is less than

4π

2

ab

n

.