Problem 34 - Algebra 2

Question

A sector is a region of a circle that is bounded by a central angle \(\theta\) and its intercepted arc. The area \(A\) of a sector with radius \(r\) and central angle \(\theta\) is given by \(A=\frac{1}{2} r^{2} \theta,\) where \(\theta\) is measured in radians. Find the area of a sector with a central angle of \(\frac{4 \pi}{3}\) radians in a circle whose radius measures 10 inches.

Step-by-Step Solution

Answer

The area of the sector is \( \frac{200\pi}{3} \) square inches, or approximately 209.44 square inches.

Step 1: Identify Known Values

From the problem, we know the radius \( r \) of the circle is 10 inches, and the central angle \( \theta \) is \( \frac{4\pi}{3} \) radians.

Step 2: Apply the Sector Area Formula

We use the formula for the area of a sector: \( A = \frac{1}{2} r^2 \theta \). Substitute the known values: \( r = 10 \) inches and \( \theta = \frac{4\pi}{3} \) radians into this formula.

Step 3: Perform Substitution

Replace \( r \) and \( \theta \) with their values in the formula: \[ A = \frac{1}{2} (10)^2 \left(\frac{4\pi}{3}\right) \].

Step 4: Calculate \( r^2 \)

Calculate \( 10^2 \), which equals 100. The expression now becomes \[ A = \frac{1}{2} \times 100 \times \frac{4\pi}{3} \].

Step 5: Simplify the Expression

First, simplify \( \frac{1}{2} \times 100 = 50 \). Now the expression is \( 50 \times \frac{4\pi}{3} \).

Step 6: Multiply the Terms

Compute the multiplication: \( 50 \times \frac{4\pi}{3} = \frac{200\pi}{3} \).

Step 7: Final Calculation or Approximation

The area \( A \) is \( \frac{200\pi}{3} \). If an approximate decimal answer is needed, substitute \( \pi \approx 3.1416 \) to find \( A \approx \frac{200 \times 3.1416}{3} \approx 209.44 \) square inches.

Key Concepts

Circle Geometry

Circle geometry is an essential branch of mathematics that explores various properties and shapes derived from a circle. A circle itself is defined as the set of all points that are equidistant from a fixed point, known as the center. The distance from the center to any point on the circle is called the radius.
Circle geometry examines components such as arcs, chords, tangents, and especially sectors, which are portions of the circle enclosed by two radii and an arc.

• Sectors can be compared to a pizza slice or a pie piece.
• Other parts, like the central angle and radius, form the sector.
• Understanding these elements helps in solving problems related to the circle, such as finding areas, lengths, and angles.

Central Angle

The central angle is pivotal in circle geometry, especially when calculating the sector's area. It is the angle formed at the circle's center by two radii, which encompass the sector.

• The central angle determines the size of the sector, with larger angles creating bigger sectors.
• Central angles are typically measured in degrees or radians, influencing how calculations are performed.
• In the context of the sector's area, the angle is crucial because it differentiates sectors from each other despite sharing the same radius.

In our example, the central angle is \(\frac{4\pi}{3}\) radians, providing the necessary component for calculating the sector's area.

Radians

Radians are a measure of angles used primarily in advanced mathematics like calculus and trigonometry, facilitating smoother calculations. Unlike degrees, which split a circle into 360 parts, radians define angles based on the circle's radius. One full circle is equal to \(2\pi\) radians.
Radians are crucial for efficient computation in mathematics, especially when dealing with formulas connected to geometric figures like circles.

• This method of measurement aligns naturally with circular motion and periodic functions.
• For example, \(\frac{\pi}{2}\) radians correspond to 90 degrees, a quarter of a full circle.
• The problem given uses radians to simplify the calculation of the sector's area without the conversion complications of degrees.

Geometry Formula

Geometry often involves formulas that distill complex relationships into usable mathematical expressions. For sectors, the area formula \(A = \frac{1}{2} r^2 \theta\) calculates the space within a sector based on its radius \(r\) and central angle \(\theta\).

• This formula highlights the proportional relationship between the radius, angle, and sector size.
• Using radians for \(\theta\) ensures direct application without additional conversion.
• The formula integrates both linear and angular components, showcasing the interplay needed to navigate closed geometries like circles effectively.

In practice, understanding this formula enables us to evaluate circular portions quickly and effectively, making it a powerful tool for problem-solving involving sectors and arced regions.