Problem 89 - Precalculus

Question

Find the area of the circular sector given the indicated radius and central angle. Round your answers to three significant digits. $$\theta=\frac{3 \pi}{8}, r=2.2 \mathrm{km}$$

Step-by-Step Solution

Answer

The area of the sector is approximately 2.85 km².

Step 1: Understanding the Problem

We need to find the area of a sector of a circle. A sector is defined by a radius and a central angle. Given are the central angle \( \theta = \frac{3\pi}{8} \) and the radius \( r = 2.2 \text{ km} \). The formula for the area \( A \) of a sector is \( A = \frac{1}{2} r^2 \theta \) when \( \theta \) is in radians.

Step 2: Applying the Formula

Let's apply the formula for the area of a sector. Substitute the given radius \( r = 2.2 \) km and angle \( \theta = \frac{3\pi}{8} \) into the formula: \[ A = \frac{1}{2} (2.2)^2 \left(\frac{3\pi}{8}\right) \].

Step 3: Calculating the Squared Radius

Calculate \((2.2)^2\). \[ (2.2)^2 = 4.84 \].

Step 4: Computation of Area

Now substitute the calculated squared radius into the area formula: \[ A = \frac{1}{2} \times 4.84 \times \frac{3\pi}{8} \]. First, calculate \( \frac{3\times4.84\times\pi}{8} \). This equals approximately 5.702. Divide by 2 to get:\[ A \approx \frac{5.702}{2} = 2.851 \text{ km}^2 \].

Step 5: Rounding the Answer

Finally, round the answer to three significant digits. The area of the sector is \( 2.85 \text{ km}^2 \) (rounded to three significant digits).

Key Concepts

Circular Sector

A circular sector is a portion of a circle, resembling a "slice of pie," that is enclosed by two radii and an arc. It is an important concept in geometry, particularly when dealing with circle measurements.
An easy way to think of a sector is by visualizing it as a part of the circle's area that is bordered by the circle's circumference and two straight lines extending from the circle's center. The area of a sector is a fraction of the entire area of the circle. This fraction is determined by the size of the central angle – the angle between the two radii.

• To calculate the area of a circle, you need the radius and the central angle.
• The formula for the area of a sector is given by \( A = \frac{1}{2} r^2 \theta\) where \( r\) is the radius and \( \theta\) is the angle in radians.

Understanding circular sectors is crucial for tasks such as calculating areas of pie charts, portions of fields, or curved structures.

Central Angle

The central angle is the angle formed by two radii of a circle with its vertex at the center of the circle. It is measured in degrees or radians and determines how large the sector of the circle will be.
A larger central angle implies a larger sector of the circle, while a smaller angle results in a smaller sector. An important characteristic to note about central angles in a circle is that they sum up to 360 degrees if measured in degree units or \( 2\pi\) if in radians.

• In the context of sectors, the central angle specifically informs the percentage of the circle's complete area represented by the sector.
• The size of the sector is directly proportional to the central angle measurement.

For the given exercise, the known central angle was \(\theta = \frac{3\pi}{8} \), facilitating the calculation of the sector's area.

Radian Measure

Radians are a unit of angular measurement used in the circle's context, crucial for calculating sector areas. One radian is the angle created when the arc length equals the radius of the circle.
Using radians as a unit eases mathematical operations in geometry, such as integrating the angle into formulas for areas or arc lengths, without the need for conversion factors as with degrees.

• The conversion between the usual degree measure and radians is \(180^ ext{o} = \pi\) radians.
• This makes a full circle \(2\pi\) radians and a half circle \(\pi\) radians.

In solving the original problem, the angle given as \(\frac{3\pi}{8} \) radians allowed direct substitution into the sector area formula.

Geometry

Geometry is a branch of mathematics dealing with shapes, sizes, and the properties of space. Circular sectors are among the essential geometric figures studied under this discipline.
Understanding the calculation and properties of a sector is part of comprehending the broader geometry of circles, which includes various formulas for determining perimeter, area, and lengths of arcs.

• Sectors help in real-world applications such as architecture, engineering, and data visualization, where pie charts and arcs are used.
• Geometry provides the tools to convert theoretical concepts into practical solutions.

The problem demonstrates geometry fundamentals by applying the formula for sector area, an essential aspect of circle geometry, for practical exercises.