Textbook Question
Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5.
r = 40.0 mi, θ = 135°
Ch. 3 - Radian Measure and The Unit Circle
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 4, Problem 53
Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5.
r = 12.7 cm, θ = 81°
Verified step by step guidance1
Recall the formula for the area of a sector of a circle: \(\text{Area} = \frac{\theta}{360} \times \pi r^{2}\), where \(\theta\) is the central angle in degrees and \(r\) is the radius.
Identify the given values: radius \(r = 12.7\) cm and central angle \(\theta = 81^\circ\).
Substitute the given values into the formula: \(\text{Area} = \frac{81}{360} \times \pi \times (12.7)^{2}\).
Calculate the square of the radius: \((12.7)^{2}\), then multiply by \(\pi\) and the fraction \(\frac{81}{360}\).
After performing the multiplication, round the result to the nearest tenth to express the final area.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Area of a Sector
The area of a sector of a circle is the portion of the circle's area enclosed by two radii and the arc between them. It is calculated using the formula (θ/360) × π × r² when θ is in degrees, where r is the radius and θ is the central angle.
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Central Angle in Degrees
The central angle θ is the angle formed at the center of the circle by two radii. It determines the size of the sector and must be expressed in degrees or radians to use the appropriate area formula.
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Rounding and Precision
After calculating the area, the result should be rounded to the nearest tenth as specified. This involves understanding decimal places and applying proper rounding rules to present the final answer accurately.
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