Ellipse (please help)

Let the eqn. of the ellipse be x² / a² + y² / b² = 1

2 a = 6 cm { Given
2 b = 4 cm { Given

Hence eccentricity e = √1 - b ² / a² = √5 / 9 = √5 / 3

Clearly , the vetices of the elipse are ( -3 ,0 ) and ( 3 , 0) . Hence , the effective length of the string should be 6 cm so that it covers the max. distance of 6 চm between the 2 vertices .

And the dist. between the focis =dist. between the nails = 2 a e = 6 x √5 / 3 = 2 √5

How can the perimeter of ellipse be equal to 2a..?