Originally Posted by

**HallsofIvy** You say you "have the formula" so I presume you know that the arclength is given by $\displaystyle \int \sqrt{r^2+ (dr/d\theta)^2} d\theta$.

Here, $\displaystyle r= a cos^2(\theta/2)$ so $\displaystyle dr/d\theta= -a sin(\theta/2)cos(\theta)$ and so $\displaystyle r^2+ (dr/d\theta)^2= a^2 cos^4(\theta/2)+ a^2 sin^2(\theta/2)cos^2(\theta/2)$$\displaystyle = a^2 cos^2(\theta/2)(cos^2(\theta/2)+ sin^2(\theta/2))= a^2 cos^2(\theta/2)$.

I don't see anything difficult about that!