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 = 29.2 m, θ = 5π/6 radians
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1. Measuring Angles
Radians
Problem 55
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°
Verified step by step guidance1
Recall the formula for the area of a sector of a circle: \(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
Identify the given values: radius \(r = 40.0\) miles and central angle \(\theta = 135^\circ\).
Substitute the given values into the formula: \(\text{Area} = \frac{135}{360} \times \pi \times (40.0)^2\).
Simplify the fraction \(\frac{135}{360}\) by dividing numerator and denominator by their greatest common divisor to make calculations easier.
Calculate the numerical value of the area by multiplying the simplified fraction, \(\pi\), and the square of the radius, then round the result to the nearest tenth.
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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, where θ is the central angle in degrees and r is the radius. This formula helps find the fraction of the circle's area corresponding to the sector.
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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. When given in degrees, it must be used directly in the sector area formula as a fraction of 360°, representing the full circle.
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Rounding and Units
After calculating the area, the result should be rounded to the nearest tenth to match the problem's requirement. Additionally, the units of the area will be the square of the radius units (e.g., square miles if radius is in miles), which is important for interpreting the final answer correctly.
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