In geometry, how is the radius of a circle defined?

The radius of a circle is defined as the distance from the center of the circle to any point on its circumference. This definition is fundamental in geometry because the radius is critical in determining various properties of the circle, including calculating its area and circumference.

By focusing on the relationship between the center and points on the circumference, the radius helps establish the basic structure of the circle itself. The radius also plays a significant role in many geometric formulas, allowing for the definition of circles in mathematical contexts.

Other options, while relating to circles, do not accurately define the radius: the distance across the circle refers to the diameter, which is twice the length of the radius; the length of a line segment connecting two points on the circle relates to chords; and the mention of a formula for calculating area refers to the area of the circle rather than the radius itself. Thus, the correct understanding of the radius directly aligns with the definition provided.