How is the arc length (s) of a circle related to the radius (r) and central angle (θ)?

The arc length (s) of a circle is directly proportional to both the radius (r) and the measure of the central angle (θ) in radians. The relationship is expressed by the formula ( s = rθ ), which indicates that the longer the radius or the larger the central angle, the greater the length of the arc along the circumference of the circle.

When considering this relationship, if you increase the radius of the circle while keeping the angle the same, the arc length also increases proportionally. Likewise, if the radius remains constant and the angle increases, the arc length will similarly increase. This makes the formula ( s = rθ ) a direct and straightforward way to calculate arc lengths in circular geometry.

In contrast, other formulations presented do not correctly represent the relationship between these variables. The equation that relates the arc length, central angle, and radius specifically utilizes a multiplication of the radius by the angle in radians, hence supporting the understanding that the arc length is indeed dependent on both dimensions linearly rather than through any squared or reciprocal relationships.