Problem 121 - A Graphical Approach to Precalculus with Limits

Question

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) \(r=12.7\) centimeters; \(\theta=81.0^{\circ}\)

Step-by-Step Solution

Answer

The area of the sector is approximately 113.947 square centimeters.

Step 1: Convert the Angle to Radians

To find the area of a sector, the angle should be in radians. Use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \) to convert the angle: \[\theta = 81^{\circ} \times \frac{\pi}{180} = \frac{81\pi}{180} \approx 1.4137\,\text{radians}\]

Step 2: Use the Formula for the Area of a Sector

The area \(A\) of a sector is given by \( A = \frac{1}{2}r^2 \theta \). Now, substitute the given values \( r = 12.7\, \text{cm} \) and \( \theta \approx 1.4137\, \text{radians}\): \[A = \frac{1}{2} \times 12.7^2 \times 1.4137 \]

Step 3: Calculate the Area

Perform the calculation using the values obtained: \[A = \frac{1}{2} \times 161.29 \times 1.4137 \approx 113.947\,\text{square centimeters}\]

Key Concepts

Radians Conversion

When calculating the area of a sector, it's crucial to express the central angle in radians instead of degrees. This might seem trivial but is essential for consistency with the formula. To convert degrees to radians, you can use the conversion formula:

• Radians = Degrees × \(\frac{\pi}{180}\)

This formula stems from the fact that the circumference of a full circle is \(2\pi\) radians, which matches 360 degrees. Thus, dividing 360 degrees by \(2\pi\) gives the conversion factor of \(\frac{\pi}{180}\). For example, if we have an angle of \(81^{\circ}\), converting it to radians is straightforward: \[ \theta = 81^{\circ} \times \frac{\pi}{180} \] This becomes approximately \(1.4137\) radians when evaluated. Always ensure your final angle measure is in radians to apply formulas directly without any mismatch.

Central Angle

The central angle of a sector is a measure from the circle's center, dictating the size of the sector. Think of it as the pivot determining how big a 'slice' of the circle will be. Measured in radians for formulas, it can initially be in degrees, necessitating conversion as seen previously. The central angle's role is pivotal because it directly affects the sector's area calculation. The formula for the area of a sector combines the radius of the circle, the central angle, and a constant factor. Given by: \[ A = \frac{1}{2}r^2\theta \] Here, \(\theta\) is the central angle in radians, and \(r\) represents the circle's radius. If this angle increases, the sector captures more of the circle's complete 360-degree, or \(2\pi\) radian, total. An angle · area correlation exists where a larger angle results in a proportional increase in the sector's area.

Circle Geometry

The geometry of a circle revolves around a few key components, such as the radius, diameter, and central angle, which together define the circle’s sections or sectors. A sector resembles a "pizza slice," complete with an arc and a straight angle from the circle's center. Understanding these parts is crucial in mastering circle geometry.

• Radius: This is the line from the center to any point on the circle's perimeter. It is a constant factor in calculations involving circle areas and sectors.
• Diameter: Twice the radius, spanning across the circle through its center.
• Circumference: The total distance around the circle, calculated as \(2\pi\times \text{radius}\).

For the sector of a circle, as we solve in the original exercise, the blended concepts of both the circle's radius and the central angle help derive its area. The sector size will change relative to the circle's total area, emphasizing different circle parts' interconnectedness. Understanding how these elements interact provides a better grasp of geometry problems involving sectors.