Problem 63 - Algebra and Trigonometry

Question

The area of a sector of a circle with a central angle of 2 rad is 16 \(\mathrm{m}^{2} .\) Find the radius of the circle.

Step-by-Step Solution

Answer

The radius of the circle is 4 meters.

Step 1: Understand the Problem

We are given the area of a sector of a circle and the central angle of the sector. Our task is to find the radius of the circle.

Step 2: Recall the Formula for Area of a Circle Sector

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

Step 3: Substitute Known Values into the Formula

We know that the area of the sector \( A = 16 \) m\(^2\) and the central angle \( \theta = 2 \) radians. Substitute these into the formula: \( 16 = \frac{1}{2} r^2 \times 2 \).

Step 4: Simplify the Equation

Simplify the equation: \( 16 = r^2 \). This is done by canceling out the \( \frac{1}{2} \times 2 \) from the right side of the equation.

Step 5: Solve for Radius

Solve for \( r \) by taking the square root of both sides: \( r = \sqrt{16} \).

Step 6: Calculate the Radius

Calculate \( \sqrt{16} \) to find that \( r = 4 \).

Key Concepts

Circle Sector

A circle sector is a portion of a circle enclosed by two radii and the connecting arc between them. Imagine a pizza slice; that's a perfect representation of a sector. The size of a sector is determined by its central angle and the radius of the circle.
To describe a circle sector:

• The two straight lines (radii) start at the center and meet the circle at different points, forming the boundaries.
• The curved line (arc) between these radii completes the sector.

Differentiating the sector from a full circle is crucial in solving problems related to areas and angles. The specific formula used for calculating the area of a sector makes it simple to find missing values when two variables are known.

Circle Radius

The radius of a circle is a line segment from the center of the circle to any point on its perimeter. It is crucial in defining the size and scale of a circle. When solving for things like the area of a circle's sector, the radius is a key component.
Important aspects of circle radius include:

• All radii in a single circle are equal in length.
• The radius is half the length of the diameter (total distance across the circle through the center).

This consistent trait of equal radii makes calculating different properties of circles much more straightforward. For example, knowing the radius allows for simple calculations of circumference and area.

Central Angle

A central angle in a circle is an angle whose vertex is the center of the circle, and whose sides (rays) are the radii that meet the circle's circumference. It's expressed in degrees or radians and directly determines the arc length and sector size.
The attributes of central angles include:

• Various units: often measured in radians when involved in sector calculations.
• Forms a complete circle when it's 360 degrees or 2π radians.

The central angle's role in finding a sector's area is paramount, as this determines the exact portion of the circle that is considered in calculations. Its measurement is crucial for determining how much of the circle’s surface area is included.

Area of a Sector

The area of a sector of a circle refers to the amount of space enclosed by the sector. It's a fraction of the entire circle's area, dictated by the central angle and the circle's radius. This segment of geometry provides vital applications in various mathematical and real-world contexts.
To find the area of a sector, the formula is:
\[ A = \frac{1}{2} r^2 \theta \]
Where:

• \( A \) represents the area of the sector,\( r \) is the radius of the circle, and\( \theta \) is the central angle in radians.

In practical exercises, understanding each component of this formula allows you to solve for missing elements, such as determining the circle's radius when the area and angle are known. This formula connects directly to a consistent method for handling circular segment problems.