Which mathematical discipline is best used to determine the relationship between the radius of a circle and its area?

The mathematical discipline that is best suited for determining the relationship between the radius of a circle and its area is geometry. Geometry directly deals with the properties and measurements of shapes, including circles.

In this context, the area of a circle can be calculated using the formula (A = \pi r^2), where (A) is the area and (r) is the radius. This formula illustrates the geometric relationship between the radius and the area, showing that the area increases with the square of the radius.

Algebra, while it deals with symbols and the manipulation of equations, is not the primary focus when considering the specific geometric relationship of shapes like circles. Trigonometry primarily deals with the relationships between angles and sides in triangles, and calculus focuses on rates of change and the accumulation of quantities, neither of which is necessary for simply determining the area of a circle based on its radius. Thus, geometry is the most appropriate mathematical discipline for this specific task.