Having decided that making a JKish style of cabinet with a coopered door is a 'good thing' I consulted the 'worthies' on UKWorkshop with a view to solving the problem of the cooperage angle.
It transpires that the sagitta (or amount of 'bendiness' accross the chord of an arc) can be calculated quite easily using some data in the equation, but the thing that I don't know is the radius of the arc. Happily however, there's a second equation immediatly underneath on the same page showing how to calcualte it, so with a combination of the two I ought to be able to sort out my little problem.
One other solution was provided by Richard Jones when he said... 'that you are dealing with a circle, so the calculations should be relatively straightforward. The profile of the door looking down from the top is an arc, and joining the outer edges of the door to the centre of the circle forms a segment. All you need to do is find out how many degrees the two radius lines form where they meet at the circle centre and you are on the way. If this angle is, for example, 20º then the complementary angle is 340º giving 360º when added together, the total number of degrees in a circle. You want 5 coopered pieces in your 20º arc. Therefore 360º/20 = 18, ie, you can fit 18 segments with 20º arcs into the 360º that form a circle. Next calculate 18 (segments) X 5 (coopered parts) = 90 coopered parts, effectively 90 segments. Then calculate 360º/90 = 4º, the angle each of those 90 segment forms at the circle centre. Next using triangles, the sum of all three inside corners of a triangle add up to 180º. You are dealing with an isosceles triangle so calculate 180º - 4º = 176º. Then 176º/2 = 88º, the angle formed between the outside face of each coopered part and the two bevelled edges. Set your rip saw bevel angle to this, or the complementary 2º depending on your saws protractor, or whatever your own calculations indicate going from your actual plans/drawings.'
Richard's solution still means that the radius of the circle needs to be found, which is easy given the formula above.
What's even easier is that once you've found the angle at the centre of the circle, simply divide it by the number of joints in the door (in this case 4) and then divide that number by 2 (as two pieces of wood will make one joint)
Simple always works for me...