What is the formula for the area of a circle?

The formula for the area of a circle is correctly represented as Area = πr². In this formula, 'r' stands for the radius of the circle, which is the distance from the center of the circle to any point on its circumference.

This formula originates from integral calculus and geometrical principles that relate the shape of a circle to the concept of limits in the approach of calculating areas of curved shapes. When you square the radius (r²) and multiply it by π (approximately 3.14159), you obtain the area of the circle. This relationship indicates that as the radius increases, the area grows proportionally to the square of the radius, emphasizing how significantly larger the area becomes with even small changes in the radius.

In contrast, the other selections provide incorrect formulas or concepts that do not accurately represent the area of a circle. For example, linear measurements like diameter or circumference involved in those options do not yield an area measurement when incorrectly applied in this context. Thus, the area of a circle is fundamentally defined as π times the radius squared.