Arc Length Formula Explanation

The length of an arc ($L$) in a circle is proportional to the angle ($\theta$) it subtends at the center. When the angle $\theta$ is measured in radians, the relationship is direct.

Deriving Arc Length

The formula connecting the arc length ($L$), radius ($r$), and the central angle in radians ($\theta$) is:

$ L = r\theta $

This formula represents the length of the arc corresponding to the angle $\theta$. For a minor arc, $\theta$ is typically less than $\pi$ radians.

Analyzing the Options

• Option 1: $r\theta$ - This directly matches the formula for arc length when the angle is given in radians.
• Option 2: $\frac{1}{2r^{2}\theta}$ - This expression does not correspond to any standard geometric formula for arc length.
• Option 3: $2r\sin\frac{\theta}{2}$ - This formula calculates the length of the chord connecting the endpoints of the arc, not the arc length itself.
• Option 4: $2\pi r$ - This is the formula for the circumference of the entire circle, which corresponds to a central angle of $2\pi$ radians.

Based on the standard formula, $r\theta$ is the correct expression for the length of the minor arc.