What is the formula for the circumference of a circle with radius r?

The correct formula for the circumference of a circle with radius ( r ) is expressed as ( C = 2\pi r ). This formula derives from the relationship between the circumference and the radius of the circle. In geometry, the circumference is the distance around the edge of the circle. Since the relation ( \pi ) (pi) relates the diameter of a circle to its circumference, and the diameter ( d ) is double the radius ( r ) (i.e., ( d = 2r )), substituting this into the formula ( C = \pi d ) leads us back to ( C = 2\pi r ).

Understanding this is crucial because it highlights the fundamental geometry concepts that connect the radius, diameter, and circumference, thereby reinforcing the relationships between these elements of a circle. This formula is widely used in various calculations involving circles and helps establish a foundational understanding of circular measurements.