Problem 364 - Precalculus

Question

Find the area of the sector with radius of 8 feet and an angle of \(\frac{5 \pi}{4}\) radians.

Step-by-Step Solution

Answer

The area of the sector is \( 40\pi \) square feet.

Step 1: Understand the Formula for Sector Area

To find the area of a sector, use the formula: \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians.

Step 2: Identify the Given Values

From the problem, we know that the radius \( r = 8 \) feet and the angle \( \theta = \frac{5\pi}{4} \) radians.

Step 3: Substitute the Values into the Formula

Substitute \( r = 8 \) and \( \theta = \frac{5\pi}{4} \) into the area of sector formula: \( A = \frac{1}{2} \times 8^2 \times \frac{5\pi}{4} \).

Step 4: Simplify the Expression

First calculate \( 8^2 = 64 \). Substitute back to get: \[ A = \frac{1}{2} \times 64 \times \frac{5\pi}{4} \].

Step 5: Calculate the Area

Compute \( \frac{1}{2} \times 64 = 32 \), then multiply by \( \frac{5\pi}{4} \): \[ A = 32 \times \frac{5\pi}{4} = 32 \times \frac{5\pi}{4} = 40\pi \].

Step 6: Final Units and Conclusion

The area of the sector is \( 40\pi \) square feet, as the result must include the appropriate unit 'square feet' for the area.

Key Concepts

Understanding Radius

In geometry, the "radius" refers to the distance from the center of a circle to any point on its circumference. This is a crucial component when calculating different properties of a circle, including the area and the perimeter.

• Symbol: Typically denoted by the letter \( r \).
• Key Role: In the context of a sector, it helps determine the size of the sector's area.
• Example: For a radius \( r = 8 \) feet, this value is used to find various circle properties.

The radius is always measured in linear units, like feet or meters, depending on problem settings. Understanding the radius is fundamental as it directly affects the calculations for circles.

Central Angle Explained

The "central angle" of a circle is the angle subtended at the center of the circle by two radii. This angle is essential for understanding how much of the circle a sector occupies.

• Measurement: It is often given in radians, though it can also be expressed in degrees.
• Notable Point: A full circle is \( 2 \pi \) radians or 360 degrees.
• Sector Calculation: Knowing the central angle allows us to find the area of that specific sector of a circle.

In our example, the central angle \( \theta \) is \( \frac{5\pi}{4} \) radians, indicating a sizable portion of the circle.

Radians and Their Importance

"Radians" are a unit of angular measurement used in many areas of mathematics and science, particularly in trigonometry and calculus. Understanding radians is crucial for working with angular measurements in formulas and equations.

• Definition: One radian is the angle at the center of a circle that makes the arc length equal to the radius.
• Relation to Degrees: 180 degrees is equivalent to \( \pi \) radians; hence \( \pi \) radians is half of a circle.
• Why Use Radians: Radians offer a natural way of relating angles to distances traveled along the circle's circumference.

In our specific problem, the angle given is in radians, simplifying the formula for the sector's area.

Calculating the Area of a Circle

The "area of a circle" is a measure of the space enclosed by the circle's circumference. It's a vital concept in understanding how circles and sectors work.

• Formula: The area \( A \) of a circle can be calculated with the formula \( A = \pi r^2 \).
• Why It Matters: Knowing the area is key for comparisons and conversions, such as finding the fraction of the circle that corresponds to a given sector.
• Application: The area of a sector is derived using both the circle's area and the central angle (in radians), showcasing how these concepts interlink.

To find the area of a sector of a circle, we apply a proportion of the circle's area based on the central angle, which is expressed in radians.