Denote by [tex]D[/tex] the unit disk centered at the origin,
[tex]D=\{(x,y)\mid x^2+y^2\le1\}[/tex]
which has area [tex]\pi[/tex]. Then [tex]X,Y[/tex] have joint PDF
[tex]f_{X,Y}(x,y)=\begin{cases}\frac1\pi&\text{for }(x,y)\in D\\0&\text{otherwise}\end{cases}[/tex]
Transform the pair of random variables [tex]X,Y[/tex] to a new pair [tex]R,\Theta[/tex] such that
[tex]\begin{cases}X=R\cos\Theta\\Y=R\sin\Theta\end{cases}[/tex]
In this new coordinate system, the support is
[tex]D=\{(r,\theta)\mid0\le r\le1,0\le\theta\le2\pi\}[/tex]
Using the method of transformations, the joint PDF of [tex]R,\Theta[/tex] is
[tex]f_{R,\Theta}(r,\theta)=f_{X,Y}(r\cos\theta,r\sin\theta)|\det J|[/tex]
where [tex]J[/tex] is the Jacobian matrix for the transformation. We have
[tex]J=\begin{bmatrix}\dfrac{\partial x}{\partial r}&\dfrac{\partial x}{\partial\theta}\\\\\dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial\theta}\end{bmatrix}=\begin{bmatrix}\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\end{bmatrix}\implies|\det J|=|r\cos^2\theta+r\sin^2\theta|=r[/tex]
Then
[tex]f_{R,\Theta}(r,\theta)=\begin{cases}\frac r\pi&\text{for }0\le r\le1,0\le\theta\le2\pi\\0&\text{otherwise}\end{cases}[/tex]
Since the joint density doesn't depend on [tex]\theta[/tex] (it varies uniformly over the interval [tex][0,2\pi][/tex] with probability [tex]\frac1{2\pi}[/tex]), it follows that [tex]R[/tex] and [tex]\Theta[/tex] are independent of one another and
[tex]f_R(r)=\begin{cases}2r&\text{for }0\le r\le1\\0&\text{otherwise}\end{cases}[/tex]
Then letting [tex]S=R^2[/tex], by the method of transformations we have
[tex]f_S(s)=f_R(\sqrt s)\left|\dfrac{\mathrm dr}{\mathrm ds}\right|=\begin{cases}1&\text{for }0\le s\le1\\0&\text{otherwise}\end{cases}[/tex]
