Problem 104 - Precalculus : Building Concepts and Connections

Question

Consider an angle \(\theta\) in standard position whose vertex coincides with the center of a circle of radius \(r .\) The portion of the circle bounded by the initial side and the terminal side of the angle \(\theta\) is called a sector of the circle. (a) If \(A\) is the area of the circle, then \(A_{s}=A \frac{\theta}{2 \pi}\) represents the area of the sector because \(\frac{\theta}{2 \pi}\) gives the fraction of the area covered by the sector. Show that the area of a sector, \(A_{s},\) is \(A_{s}=\frac{r^{2} \theta}{2} .\) Here theta is in radians. (b) Find \(A_{s}\) if \(\theta=\frac{\pi}{3}\) and \(r=12\) inches.

Step-by-Step Solution

Answer

The area of the sector, \(A_s\), is \(24\pi \, \text{sq. inches}\).

Step 1: Understand the formulaic relationship

The area of a whole circle is defined as \(A = \pi r^2\) and it covers an angle of \(2\pi\) radians. Therefore, the area covered by any angle will be its ratio to \(2\pi\) times the total area. This means that \(A_s = A \frac{\theta}{2\pi}\). Substituting \(A\) with \(\pi r^2\), we get \(A_s = \pi r^2 \frac{\theta}{2\pi}\)

Step 2: Simplify the relationship

Simplify the above equation by canceling out \(\pi\) on both sides to get \(A_s = \frac{r^2 \theta}{2}\). This is the formula for the area of a sector of a circle given the radius and the central angle in radians.

Step 3: Substituting values

Now substitute given values of \(\theta = \frac{\pi}{3}\) and \(r=12\) inches into the equation \(A_s = \frac{r^2 \theta}{2}\)

Step 4: Calculate

Do the math to determine the area: \(A_s = \dfrac{(12^\text{in})^2 \times \frac{\pi}{3}}{2} = 24\pi \, \text{sq. inches}\).

Key Concepts

Understanding the Radius of a Circle

The radius of a circle is a key measurement that helps us define other important characteristics of the circle. Think of it as the distance from the center of the circle to any point along its edge. This measurement is consistent throughout the circle, meaning it's the same no matter where you measure it from the center. It is denoted by the symbol \( r \).

• The radius is essential when determining the area, circumference, or characteristics of any part of the circle.
• When working with circle sectors, the radius becomes an integral part of calculating the sector's area.

This is because the formula for the area of a sector involves squaring the radius. The squared radius \( r^2 \) forms the basis of the operations within these calculations, making it a foundational aspect of any circle-related problem.

Defining Angle in Radians

Radians offer a way of measuring angles that is particularly useful in calculus and when dealing with circles. Unlike degrees, radians relate directly to the arc's length and the circle's radius. There are \( 2\pi \) radians in a full circle, which equals 360 degrees. This makes one radian approximately 57.3 degrees.

• Using radians simplifies many formulas, especially those involving circle sectors.
• Radians enable a direct link between the angle and the distance traveled along the circle's circumference.

Understanding radians is crucial for working out the area of a sector. In our problem, using radians (\( \theta \)) ensures we can directly correlate an angle to a segment of the circle's entire circumference, without needing further adjustments like converting from degrees.

Calculating the Area of a Sector

The area of a sector is a proportional part of the area of the whole circle. Essentially, it's the "slice" of the circle defined by the central angle \( \theta \) in radians and the radius \( r \). The formula for the area of a sector is \( A_s = \frac{r^2 \theta}{2} \).
Here's how the formula works in steps:

• The full circle area is \( \pi r^2 \), covering the angle \( 2\pi \).
• The sector's area then becomes \( \frac{r^2 \theta}{2} \), derived by taking \( r^2 \) and multiplying it by \( \theta/2 \).

This formula allows you to calculate the exact area of any slice of a circle, based on its central angle and radius. When \( \theta = \frac{\pi}{3} \) and \( r = 12 \) inches, plugging in these values gives the precise area of the sector as \( 24\pi \) square inches, demonstrating how angular measurement directly impacts area calculations.