Fitting text to "curved pie shape"

I want to achieve an effect as if text is filling the segments of a dartboard.  So the text path would be curved and fill the segment running towards the centre.  If you picture the black 20 segement of a dartboard the text should fit that shape but follow the curve. The image shows the sort of effect I want....  Ideally each section should contain the "paragraph" of text rather than it running as if around the grooves of a record.

I am using InDesign CS6

I know how to fill a shape in InDesign.  However the text would be in straight lines rather than following the curves of the pie shape.

Does anyone know if what I want is possible with InDesign and how.  If it is not, what is the way to do it on, say Illustrator?

Thank you for any help!