**This is an article from my personal notebook few years back. Very few read it. As I found this is a right place so thought of sharing it here **
One of the important properties of a circle is that it has minimum boundary length with same area among all possible shapes. We can derive that with little imagination. Suppose in the beginning, an irregular area in a flat surface consists of tiny little frictionless particles with infinitesimally small areas. The flat surface is perfectly frictionless and particles are free to move in that flat surface only in any direction. The particles also possess the property of gravity. That means they attract each other by gravitational force. There is no other force acting on these particles which has got a component in the flat surface. As the particles attract to each other they will try to squeeze inside the area. This is also true for the particles at the boundary. As particles at the boundary will try to squeeze inside, this will ensure at the end that minimum possible number of particles remain at the boundary, which will create minimum possible length of the boundary. Now at the end all the particles at the boundary will experience same resultant force inward, which means to the direction perpendicular to the direction of the tangent to the boundary at that point. So that no points at the boundary will be able to squeeze more into inside than the adjacent points.
If resultant force has a direction which is in the direction of tangent (either side), that will mean that particle will have movement which will have a component in that direction. We already assumed that particles are stable at the end so there will not be any movements for any particles. This indicates resultant force on any particle at the boundary will be in the direction inwards which is perpendicular to the direction of the tangent at the point to the boundary line.
This indicates the areas at both side of this perpendicular line have to be same and symmetrical to each other. As this is true for all the points at the boundary, this indicates that the total area has got infinite rotational symmetry. This indicates that the area is a circle.
As this is true for particles in two dimension, this is equally applicable for particles in three dimensions. Because of the same very reason, among all possible shapes, sphere has the least surface area with same volume. Most celestial objects are spherical or almost spherical in nature because of the same reason.
p.s. Using Calculus of variations one can deduce that Circle has the minimum circumference. But it does not tell till the end that the equation is going to be a circle. The article was inspired by observations that when water droplets (which look like circles from the top) on the smooth floor or table when connect to each other, immediately become a bigger droplet (which again look like a bigger circle from the top).