Textbook Question
Find the measure of the central angle on a circle of radius r that forms a sector with the given area.
Radius, r: 10 feet Area of the Sector, A: 25 square feet
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1. Measuring Angles
Radians
2:28 minutesProblem 9
Textbook Question
In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 6 yards Arc Length, s: 8 yards
Verified step by step guidance1
Recall the formula that relates the arc length \( s \), radius \( r \), and central angle \( \theta \) in radians: \[ s = r \times \theta \].
Identify the given values: radius \( r = 6 \) yards and arc length \( s = 8 \) yards.
Substitute the known values into the formula: \[ 8 = 6 \times \theta \].
Solve for the central angle \( \theta \) by dividing both sides of the equation by 6: \[ \theta = \frac{8}{6} \].
Simplify the fraction if possible to express \( \theta \) in radians.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radian Measure of an Angle
A radian is a unit of angular measure based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. It provides a natural way to relate angles to arc lengths without converting to degrees.
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Relationship Between Arc Length, Radius, and Central Angle
The central angle θ (in radians) of a circle is related to the arc length s and radius r by the formula θ = s / r. This formula allows you to find the angle when the arc length and radius are known, making it essential for solving problems involving circular arcs.
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Units Consistency in Trigonometric Calculations
When calculating angles and lengths in trigonometry, it is important to ensure that all measurements are in consistent units. Here, both radius and arc length are given in yards, so no unit conversion is needed, simplifying the calculation of the radian measure.
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