Geometric Analysis of a Deltoid Area Formed by The Intersection of Two Circles

ABSTRACT This study examines the region formed by the intersection of two circles from a geometric perspective. The intersection scenarios were evaluated based on whether the radii of the circles were equal or different. When the circles intersect, a line segment is formed by connecting the two points obtained, and this segment, together with the line passing through the centres of the circles, forms the basis of a rectangle with perpendicular diagonals. The study examined the relationship between the radii of the circles and the diagonal lengths of the rectangle, and in this context, a formula was obtained that gives the area of the rectangle formed in the intersection region. The analyses showed that the area of the quadrilateral can be calculated if the radii of the circles and the length of the line segment passing through the centres in the intersection region are known. Furthermore, it was determined that the area of the quadrilateral increases up to a certain point depending on the positions of the circles and then begins to decrease, and the maximum and minimum area values were determined accordingly.