Problem 122 - A Graphical Approach to Precalculus with Limits

Question

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) $$r=18.3 \text { meters; } \theta=125^{\circ}$$

Step-by-Step Solution

Answer

The area of the sector is approximately 365.0 square meters.

Step 1: Understand the Formula for the Area of a Sector

The formula to find the area of a sector with a radius \( r \) and a central angle \( \theta \) (in radians) is \( A = \frac{1}{2} r^2 \theta \). Our aim is to apply this formula after converting the angle \( \theta \) from degrees to radians.

Step 2: Convert the Central Angle to Radians

To convert the angle from degrees to radians, use the formula: \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\). Substitute \( \theta = 125^{\circ} \) into the formula to get \( \theta = 125 \times \frac{\pi}{180} \approx 2.1817 \) radians.

Step 3: Substitute Values into the Formula

Now that we have \( \theta \) in radians, substitute \( r = 18.3 \) meters and \( \theta = 2.1817 \) into the sector area formula: \( A = \frac{1}{2} \times 18.3^2 \times 2.1817 \). This simplifies to \( A \approx \frac{1}{2} \times 334.89 \times 2.1817 \approx 365.0 \) square meters.

Key Concepts

Radius and Central Angle

The radius and central angle are two fundamental components that describe a sector of a circle. The radius is the distance from the center of the circle to any point on its circumference. For this problem, the radius is given as 18.3 meters.
The central angle is the angle that defines the arc of the sector, originating from the circle's center and subtends the arc at its circumference. An important distinction here is that this angle can be expressed in degrees or radians. In this scenario, the central angle is 125 degrees.
Understanding these two parameters is essential as they determine the size of the sector and help calculate its area.

Degree to Radian Conversion

When working with the area of a sector, it's often required to convert the given angle from degrees to radians. This is crucial because most mathematical formulas involving circular measurements assume the angle is in radians.
The conversion formula is simple:

• Multiply the angle in degrees by \( \frac{\pi}{180} \)

In our example, we have a central angle of 125 degrees. We convert this to radians by using the formula:

• \( \theta_{\text{radians}} = 125 \times \frac{\pi}{180} \approx 2.1817 \)

After conversion, we can proceed with calculations using the radian measure.

Mathematical Formula Application

Once the angle is in radians, we can use the formula to calculate the area of the sector:

• \( A = \frac{1}{2} r^2 \theta \)

This formula is intuitive as it mirrors the formula for the area of a circle but adjusted for the fraction of the circle that the sector represents. The original full circle area formula, \( \pi r^2 \), scales down using \( \theta \), giving us the sector's specific area.
For our problem, the values to substitute are:

• \( r = 18.3 \) meters
• \( \theta = 2.1817 \) radians

After substitution and calculation:

• \( A = \frac{1}{2} \times (18.3)^2 \times 2.1817 \approx 365.0 \) square meters

Utilizing this formula makes calculating the sector's area straightforward and practical.

Geometry

Geometry helps us understand spatial relationships, angles, and measurements like the area of sectors. In geometry, sectors are a vital part of learning about circles, arcs, and angles.
This exercise involves several geometry concepts:

• Central Angle: Describes the portion of the circle the sector represents.
• Radius: A characteristic that defines the circle's size and consequently the sector's size.
• Sectors: Parts of the circle defined by an arc and two radii, key for understanding parts of circles.

By using geometry, we connect these components together. The study of sectors not only exemplifies understanding how to calculate areas but also how angles and radii interact to form different portions of a circle. This geometric understanding is foundational to grasping more complex mathematical concepts.