The area of a circle is calculated using the formula A = πr², where A represents the area, π (pi) is a constant approximately equal to 3.14159, and r is the radius of the circle. This formula illustrates that the area is proportional to the square of the radius, meaning that any changes to the radius significantly affect the area. Understanding this formula is crucial in various applications, including geometry, physics, and engineering, as it helps determine the space within circular shapes.