Which of the following is the radian measure of a central angle that subtends an arc equal in length to the radius of the circle?

• A. $\pi$ radians
• B. $2\pi$ radians
• C. $90$ degrees
• D. $1$ radian

Answer: (D)

Verified step by step guidance

1. Recall the definition of a radian: a radian is the measure of a central angle in a circle where the length of the arc it subtends is equal to the radius of the circle.
2. Express the relationship between the arc length \( s \), radius \( r \), and central angle \( \theta \) in radians using the formula: \[ s = r \times \theta \]
3. Since the arc length \( s \) is equal to the radius \( r \), substitute \( s = r \) into the formula: \[ r = r \times \theta \]
4. Divide both sides of the equation by \( r \) (assuming \( r \neq 0 \)) to isolate \( \theta \): \[ \theta = 1 \]
5. Conclude that the central angle that subtends an arc equal in length to the radius of the circle measures \( 1 \) radian.