The length of a curve C is given by the integral,
[tex]\displaystyle\int_C\mathrm ds[/tex]
where the line element ds is
[tex]\mathrm ds=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt[/tex]
where [tex]x=x(t)[/tex] and [tex]y=y(t)[/tex] are parameterizations of C.
In this case, we have
[tex]x(\theta)=r(\theta)\cos\theta[/tex]
[tex]y(\theta)=r(\theta)\sin\theta[/tex]
Differentiate with respect to [tex]\theta[/tex] to get
[tex]\dfrac{\mathrm dx}{\mathrm d\theta}=\dfrac{\mathrm dr}{\mathrm d\theta}\cos\theta-r(\theta)\sin\theta[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm d\theta}=\dfrac{\mathrm dr}{\mathrm d\theta}\sin\theta+r(\theta)\cos\theta[/tex]
[tex]\dfrac{\mathrm dr}{\mathrm d\theta}=\sin\left(\dfrac\theta2\right)\cos\left(\dfrac\theta2\right)=\dfrac12\sin\theta[/tex]
So the arc length is
[tex]\displaystyle\int_C\mathrm ds=\int_0^a\sqrt{\left(\dfrac12\sin\theta\cos\theta-\sin^2\left(\dfrac\theta2\right)\sin\theta\right)^2+\left(\dfrac12\sin\theta\sin\theta+\sin^2\left(\dfrac\theta2\right)\cos\theta\right)^2}\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_0^a\sqrt{\sin^2\left(\dfrac\theta2\right)}\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_0^a\sqrt{\frac{1-\cos\theta}2}\,\mathrm d\theta[/tex]
