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

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

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 and . The total length (the circumference) is given by , where is the complete elliptic integral of the second kind. If (the ellipse is a circle), the circumference is . The total area is .

a

b

4aE

1-

2

(b/a)

E

a=b

4aE(0)=4aπ/2=2πa

πab