Textbook Question
CONCEPT PREVIEW Find the exact length of each arc intercepted by the given central angle.
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1. Measuring Angles
Radians
Problem 5
Textbook Question
CONCEPT PREVIEW Find the measure of each central angle (in radians).
Verified step by step guidance1
Recall that a central angle in a circle is the angle formed at the center of the circle by two radii.
Understand that the measure of a central angle in radians is related to the arc length it subtends on the circle by the formula: \(\theta = \frac{s}{r}\), where \(\theta\) is the central angle in radians, \(s\) is the arc length, and \(r\) is the radius of the circle.
Identify the given values in the problem: the arc length \(s\) and the radius \(r\) of the circle.
Substitute the known values of \(s\) and \(r\) into the formula \(\theta = \frac{s}{r}\) to express the central angle in radians.
Simplify the fraction to find the measure of the central angle in radians.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Central Angle
A central angle is an angle whose vertex is at the center of a circle and whose sides intersect the circle, forming an arc. The measure of a central angle corresponds directly to the length of the arc it intercepts, making it fundamental in relating angles to arc lengths.
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Radian Measure
Radians measure angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. This unit connects linear and angular measurements, with 2π radians equal to 360 degrees, simplifying calculations involving circles.
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Arc Length and Angle Relationship
The arc length (s) of a circle is related to the radius (r) and central angle (θ in radians) by the formula s = rθ. Understanding this relationship allows one to find the central angle if the arc length and radius are known, which is essential for solving problems involving central angles.
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