#### Answer

$8$

#### Work Step by Step

As we are given that $r=-1+\cos \theta$
Length of the curve is: $L= \int_{0}^{2\pi} \sqrt{(-1+\cos \theta)^2+(-\sin \theta)^2} d\theta$
This gives: $L=2 \int_{0}^{2\pi} \sin (\dfrac{\theta}{2}) d\theta$
This implies that
$L=-4(-1)-(-4)=4+4=8$