Radius of curvature polar

Question

$$ρ = a \sin^3({\theta/3}) , ρ=\sqrt{(x^2+y^2)}$$

Please help me find the radius of curvature of this problem I have solved parametric and polynomial forms but i am unable to get this

Answer 1

In polar coordinates $\rho=\rho(\theta)$ the radius of curvature is given by link: $$\large R=\frac{\left(\rho^2+\left(\frac{d\rho}{d\theta}\right)^2\right)^{3/2}}{\left|\rho^2+2\left(\frac{d\rho}{d\theta}\right)^2-\rho\left(\frac{d^2\rho}{d\theta^2}\right)\right|}$$