What is the unit of angle that is equal to an angle at the center of a circle whose arc is equal in length to the radius?

The unit of angle that corresponds to an angle at the center of a circle whose arc length is equal to the radius is the radian. In the context of circular motion and geometry, a radian is defined based on the relationship between the radius of a circle and the arc length.

When an angle is measured in radians, it corresponds to the arc created on the circumference of a circle where the length of the arc equals the length of the circle's radius. Mathematically, if the radius of the circle is ( r ), and the arc length ( s ) is also ( r ), then the angle ( \theta ) in radians can be expressed as:

[

\theta = \frac{s}{r}

]

This means that when ( s = r ), the angle ( \theta ) is 1 radian. Therefore, using radians as the unit of measure allows for this direct relationship between linear dimensions (arc length and radius) and angular dimensions, which is fundamental in many areas of mathematics and physics.

Other options, such as degrees, grad, and arc, do not establish this specific relationship of being defined by arc length relative to radius, making radians the appropriate choice in this context.