Which formula calculates the area of a circle?

The formula for calculating the area of a circle is indeed represented by the expression that includes the mathematical constant pi multiplied by the radius squared, which is pi times r squared. This formula stems from integral calculus and provides the measure of the space enclosed within the boundaries of the circle.

To break this down further, the radius is the distance from the center of the circle to any point on its circumference. Squaring the radius (r^2) gives a value that, when multiplied by pi, provides a precise calculation of the area. Pi is a constant approximately equal to 3.14159, and it is essential in relating the diameter of a circle to its circumference and area.

Other formulas mentioned relate to different concepts. The expression that involves 2 times pi times r calculates the circumference, which is the distance around the circle, not the area. The formula with 2 times pi times r squared does not correspond to a standard geometric measure for circles at all. Lastly, pi times d, where d is the diameter, calculates the circumference again, using the diameter instead of the radius.

Understanding this distinction reinforces the specific geometric properties of circles, emphasizing why the area is derived through the radius squared and pi specifically.