This problem gives an opportunity to explore the properties of pdfs using the mathematics of sectors of circles.
Possible approach
There are two main parts to this problem.
The first is to understand why certain shapes are firstly valid pdfs and secondly how they satisfy the technical requirement of the question. This would benefit from a discussion approach.
The second part, calculating the maximum value, will lead students into the mathematics of sectors and segments of circles.
Key questions
What properties must a pdf have?
How would the requirements of the questions relate to a graph?
In order to obtain the maximum possible value for the case of the circle of radius $1$, what do we know about the arc?