2. 3 points on the circle ( n =3) give 4 regions What happens for the next point ? ( n =4) The number of extra regions created by each new chord is the # of new intersection points plus one Therefore the total number of extra regions created by drawing all of the chords is equal to the number of chords (i.e. the ‘one’ from above statement) added to the total # of interior pts of intersection Now just need to find a way to count the # of chords and # of interior intersection points for a given number, n , of points on the circle Each chord is determined by two points on the circle. For a certain # of points on the circle, n , how many ways can we choose two points? Each interior point of intersection is determined by four points on the circle. For a certain # of points on the circle, n , how many ways can we choose four points?
3. Therefore, the number of regions, r , in terms of the number of points on the circle, n , is given by the following formula: which simplifies to * There is one region at the start – the interior of the circle – so need to add that
