Problem 65 - 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 Formula for Sector Area

The formula for the area of a sector of a circle is given by \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius, \( \theta \) is the central angle in radians, and \( A \) is the area. Here, the given area is \( 16 \text{ m}^2 \), and the central angle \( \theta = 2 \text{ rad} \).

Step 2: Substitute the Known Values

Substitute the given values into the formula: \( 16 = \frac{1}{2} r^2 \times 2 \).

Step 3: Simplify the Equation

Simplify the equation by multiplying across: \( 16 = r^2 \).

Step 4: Solve for Radius

To find \( r \), take the square root of both sides of the equation: \( r = \sqrt{16} \).

Step 5: Calculate the Radius

The square root of 16 is 4, so \( r = 4 \text{ m} \).

Key Concepts

Central Angle

The central angle is a crucial component when calculating the area of a circle's sector. This is the angle subtended by two radii of a circle. In other words, it is the angle that the sector makes at the center of the circle. Understanding the central angle is important for determining how large the sector is in comparison to the entire circle.

For calculations of a sector's area, it is essential that the central angle is measured in radians rather than degrees. Radians provide a direct relationship between the angle and the arc length of the circle. A full circle has a central angle of \(2\pi\) radians. Knowing this, you can express the size of any part of the circle as a fraction of the full circle by using the central angle in radians.

In our problem, the central angle is provided as 2 radians. This means the sector represents a portion of the circle determined by this central angle. This is pivotal to applying the sector area formula.

Radius Calculation

The radius of a circle is the constant distance from its center to any point along its perimeter. In problems involving sector areas, calculating the radius often involves using given values in a formula manner. For our problem, to find the radius, we utilize the formula for the area of a sector.

To determine the radius from the given sector area and the central angle, the formula used is:

• Sector Area, \( A = \frac{1}{2} r^2 \theta \)

By substituting the given values into this formula, we can solve for the radius.

We start with the rewritten formula: \( 16 = \frac{1}{2} r^2 \times 2 \). Simplifying this equation, we obtain: \( 16 = r^2 \). To solve for \( r \), we take the square root of both sides to find \( r = \sqrt{16} \), which gives us a radius of 4 meters.

Thus, knowing how to properly manipulate this equation is vital in finding the radius when other sector properties are known.

Sector Formula

The formula for finding the area of a sector is derived from the relationship between the central angle, radius, and the resulting sector's area. The sector area formula is:\[ A = \frac{1}{2} r^2 \theta \]

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

This formula calculates the area by finding the fraction of the area of the full circle represented by the central angle. The full circle area, \( \pi r^2 \), is scaled by the fraction represented by \( \theta / (2\pi) \) which simplifies into the formula listed above.

In the given exercise, the area of the sector is known, as well as the central angle. You can substitute known values for these into the formula to solve for the unknown radius, as illustrated in the steps provided. This highlights the utility of the sector formula for making these kinds of calculations straightforward.

Understanding and applying this formula allows students to solve a variety of radius and area-related problems associated with circular sectors.