i) First consider the triangle where is the centre of and are the centres of 2 of the smaller circles. We know that since it is 2 radii long and so by the cosine rule (using , we get
Since the number of circles is as the triangles formed by centres of smaller circle are equal and cut the larger circle into equal sections we get
rearranging for we get
ii) The area of is and the combined area of the circles is therefore rearranging we get plugging in the expression for n we found earlier we get
After iterating we get which by plugging into our equation for n yields since has to be an integer we can conclude that plugging this back into the equation for and simplifying we get the exact value for
i) First consider the triangle