Length of an Arc of an Ellipse and the Area It Subtends

vertical axis

1

axes

arc

45.

°

length = 1.4564483

area = 1.1071487

This Demonstration computes the length of an arc of an ellipse and the area that the arc subtends. Let the semimajor and semiminor axes of the ellipse be

a

and

b

. The total length (the circumference) is given by

4aE

1-

2

(b/a)



, where

E

is the complete elliptic integral of the second kind. If

a=b

(the ellipse is a circle), the circumference is

4aE(0)=4aπ/2=2πa

. The total area is

πab

.