HomeNon-Routine MathematicsRegions Made by Circles

## Regions Made by Intersecting Circles

##### Logic, Pattern and Formula for maximum number of regions made by intersection of n circles in a plane.

One circle creates 2 regions in a plane (1^{st}region inside the circle and 2

^{nd}outside the circle) Total regions = 2

2^{nd} circle can create maximum 2 more regions in the same plane when it intersects with the 1st circle.

Total Regions = 2+2= 4

^{rd}circle can create maximum 4 more regions in the same plane when it intersects with the existing 2 circles. Total regions = 2+2+4 = 8

4^{th} circle can create maximum 4 more regions in the same plane when it intersects with the existing 3 circles.

Total regions = 2+2+4+4=14

(You must be thinking that you can create more regions by shifting one of the circles but practically that will not be possible. For instance, if we shift green circle down, we will loose region 1 and there will be no region that lies only in the green circle . We will lose regions 7 and 8 if we shift it towards right.)

Similarly, maximum number of regions made by 5 circles = 22

And maximum number of regions made by 6 circles = 32**Each of the above results fits into the pattern** –

**Total number of regions created by n circles** = **n(n-1) + 2**

or

= **n ^{2}– n + 2**