Math CentralQuandaries & Queries


Xetro, a student:


Suppose you have an irregular polygon(convex or concave) with n > 3 sides.

The question is - Find some circle that will cut(in limiting case - touch) all the sides of that polygon.
It doesn't matter how many times it cuts the side(1 or 2), it just have to cut or touch it.

How to find such a circle? or how to decide if such circle even exists?
What if those segments do not form a polygon but are some arbitrary segments ?

Really want to know how to do it................

Thanks a lot..


All irregular polygons do not have a circle that is tangent to each side, the exception is the triangle. If you were to draw a polygon at random, it is unlikely that there is a circle that has every side as a tangent. Irregular polygons do not have a centre but rather a centroid which is the arithmetic mean of each of the coordinate points. For a convex irregular polygon, the centroid lies within the polygon and for a non-convex polygon the centroid lies outside the polygon. If you find the centroid of a irregular, this would be your best bet for finding a circle that cuts all sides of a irregular. Even though it is slightly more likely that you could find a circle that can cut or be tangent to the sides a an irregular polygon, there are still polygons that it would not be possible for a circle to touch all sides. Consider the polygon below:

There is no size or position I could put a circle that would touch all sides of the polygon. You can however start with the circle and draw the irregular polygon around it.

Hope this helps,


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