CONCEPT PREVIEW Find the measure of each central angle (in radians).

Verified step by step guidance

1. Understand that a central angle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle.
2. Recall that the measure of a central angle in radians is the length of the arc divided by the radius of the circle, expressed as \( \theta = \frac{s}{r} \), where \( \theta \) is the central angle in radians, \( s \) is the arc length, and \( r \) is the radius.
3. Identify the given values for the arc length \( s \) and the radius \( r \) from the problem statement.
4. Substitute the given values into the formula \( \theta = \frac{s}{r} \) to find the measure of the central angle in radians.
5. Simplify the expression to determine the measure of the central angle in radians.

Key Concepts

Central Angle

A central angle is an angle whose vertex is at the center of a circle and whose sides (rays) extend to the circumference. The measure of a central angle is directly related to the arc length it subtends on the circle. Understanding central angles is crucial for solving problems involving circular motion and geometry.

Radians

Radians are a unit of angular measure used in mathematics, particularly in trigonometry. One radian is defined as the angle formed when the arc length is equal to the radius of the circle. This unit is essential for converting between degrees and radians and is commonly used in calculus and physics.

Arc Length

Arc length is the distance along the curved line of a circle between two points. It can be calculated using the formula: arc length = radius × central angle (in radians). Understanding arc length is important for finding the measures of central angles and for applications in circular motion.