The centroid of an area is the average location of its mass, assuming the mass is distributed uniformly throughout the area.
When a circle of diameter r is removed from a larger circle of radius r, the resulting shape is called a lune. As depicted in [Image of Lune], the lune consists of two congruent sectors.
Since the removed circle is centered on the larger circle, the remaining lune is symmetrical about a vertical axis passing through the center of both circles. Due to this symmetry, the centroid of the lune will lie on this axis.
To locate the centroid, we can divide the lune into two smaller, congruent right triangles, each with a base of r and a height of r, as illustrated in [Image of Lune with Triangles]. The centroid of each triangle is located at a distance of r/3 from the base, which is also r/3 from the center of the larger circle.
Since the lune is comprised of two such triangles, the centroid of the lune coincides with the center of the larger circle and is located at a distance of r from the center of the removed circle.