The Radius Of A Circle Is 6 Kilometers. What Is The Area Of A Sector Bounded By A 126 Arc? Give The Exact
Understanding the properties of circles and their sectors is fundamental in geometry, especially when dealing with real-world applications such as land measurement, engineering, and navigation. In this article, we will explore how to calculate the area of a sector given a circle's radius and the measure of the arc that bounds it. Specifically, we will analyze the case where the radius of the circle is 6 kilometers, and the sector is bounded by a 126-degree arc. Our goal is to determine the exact area of this sector using precise mathematical formulas.
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Understanding the Basics of Circles and Sectors
Before delving into calculations, it's essential to understand some foundational concepts related to circles and their sectors.
What Is a Circle?
A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance from the center to any point on the circle is known as the radius, denoted by 'r'.What Is a Sector of a Circle?
A sector of a circle resembles a "slice" of the circle, bounded by two radii and the arc between them. It is analogous to a pie slice and is characterized by:- The radius (r) of the circle.
- The measure of the central angle (θ), usually in degrees or radians.
- The arc length (the curved part of the sector boundary).
Key Terms in Sector Calculations
- Radius (r): The distance from the center to any point on the circle.
- Central angle (θ): The angle subtended at the center of the circle by the sector.
- Arc length (L): The length of the curved boundary of the sector.
- Area of the sector (A): The surface area enclosed by the two radii and the arc.
Given Data and Objective
Let's specify the problem clearly:
- Radius of the circle, r = 6 kilometers
- Measure of the arc (central angle), θ = 126 degrees
Objective: Calculate the exact area of the sector bounded by the 126-degree arc.
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Mathematical Formulas for Sector Area
To compute the area of a sector, we use the following formulas depending on the units of the central angle:
1. Sector Area When the Central Angle Is in Degrees
\[ A = \frac{\theta}{360} \times \pi r^2 \] Where:- \( A \) = area of the sector
- \( \theta \) = central angle in degrees
- \( r \) = radius of the circle
2. Sector Area When the Central Angle Is in Radians
\[ A = \frac{1}{2} r^2 \theta \] Where:- \( \theta \) = central angle in radians
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Step-by-Step Calculation of the Sector Area
Let's proceed step by step to find the exact area.
Step 1: Confirm the Data
- Radius, \( r = 6 \) km
- Central angle, \( \theta = 126^\circ \)
Step 2: Convert Degrees to Radians (Optional)
While we can directly use degrees in the formula, sometimes converting to radians offers more flexibility.Conversion formula:
\[
\text{Radians} = \frac{\pi}{180} \times \text{Degrees}
\]
Calculating:
\[
\theta_{radians} = \frac{\pi}{180} \times 126 = \frac{126\pi}{180} = \frac{7\pi}{10}
\]
However, since our area formula in degrees is straightforward, we'll proceed with degrees.
Step 3: Apply the Sector Area Formula
Using: \[ A = \frac{\theta}{360} \times \pi r^2 \] Plug in the known values: \[ A = \frac{126}{360} \times \pi \times (6)^2 \]Calculate each component:
- Simplify \( \frac{126}{360} \):
\frac{126}{360} = \frac{7}{20}
\]
- Compute \( r^2 \):
6^2 = 36
\]
- Now, multiply:
A = \frac{7}{20} \times \pi \times 36
\]
Simplify further:
\[
A = \frac{7}{20} \times 36 \pi
\]
\[
A = \left(\frac{7 \times 36}{20}\right) \pi
\]
Calculate numerator:
\[
7 \times 36 = 252
\]
Divide:
\[
\frac{252}{20} = 12.6
\]
Thus, the exact area:
\[
A = 12.6 \pi \text{ square kilometers}
\]
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Expressing the Exact Area of the Sector
The area of the sector bounded by a 126-degree arc in a circle with a radius of 6 km is:
\[
\boxed{
\textbf{Area} = 12.6 \pi \text{ km}^2
}
\]
This is the exact value in terms of \(\pi\). If you prefer a numerical approximation, multiply by \(\pi \approx 3.1416\):
\[
A \approx 12.6 \times 3.1416 \approx 39.56 \text{ km}^2
\]
However, for precise calculations, especially in mathematical contexts, leaving the answer in terms of \(\pi\) is preferable.
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Additional Insights and Applications
Calculating the area of a sector has numerous practical applications:
- Land surveying: Determining the size of a pie-shaped plot.
- Engineering: Designing mechanical parts with curved sections.
- Navigation and GPS: Calculating the area covered within a certain angle and radius.
- Architecture: Planning curved structures or sections of circular designs.
Understanding how to derive the area in exact terms helps in accurate planning and avoids approximations that could lead to errors in critical applications.
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Summary and Key Takeaways
- The area of a sector depends on the radius and the angle subtended at the center.
- When the angle is in degrees, the formula \(A = \frac{\theta}{360} \times \pi r^2\) is straightforward.
- For a circle with radius 6 km and a 126-degree arc, the exact sector area is \(12.6 \pi \text{ km}^2\).
- Numerical approximations provide a close estimate but are less precise than the symbolic form.
Final Remarks
Calculating the area of a sector is a fundamental skill in geometry with broad applications. By understanding the relationship between the radius, the central angle, and the sector area, you can solve various problems involving circles with confidence. Remember to always pay attention to units and conversions, especially between degrees and radians, to ensure accuracy in your calculations.
Whether you're a student, engineer, or land surveyor, mastering these formulas enhances your ability to work with circular segments precisely and efficiently.
