Problem 67 - Precalculus Mathematics for Calculus

Question

The area of a circle is \(72 \mathrm{cm}^{2} .\) Find the area of a sector of this circle that subtends a central angle of \(\pi / 6\) rad.

Step-by-Step Solution

Answer

The area of the sector is 6 cm².

Step 1: Formula for Sector Area

The formula for finding the area of a sector is given by \( A = \frac{\theta}{2\pi} \times \pi r^2 \), where \( \theta \) is the central angle in radians and \( \pi r^2 \) is the area of the full circle. In this case, \( \theta = \frac{\pi}{6} \).

Step 2: Calculate Radius

We know the total area of the circle is \( \pi r^2 = 72 \). Solving for \( r^2 \), we have \( r^2 = \frac{72}{\pi} \).

Step 3: Substitute Values

Substitute \( \theta = \frac{\pi}{6} \) and \( r^2 = \frac{72}{\pi} \) into the sector area formula: \( A = \frac{\frac{\pi}{6}}{2\pi} \times 72 \).

Step 4: Simplify the Expression

Simplify the expression: \( A = \frac{1}{12} \times 72 = 6 \). The area of the sector is \( 6 \mathrm{cm}^2 \).

Key Concepts

Circle Area Formula

The formula to calculate the area of an entire circle is a fundamental concept in geometry. It helps us understand a circle's size when given its radius. The area of a circle is determined using the formula:

• \( A = \pi r^2 \)

Here, \( A \) represents the area, \( \pi \approx 3.14159 \) is a constant, and \( r \) is the circle's radius. To find an area, you simply square the radius (multiply it by itself) and then multiply by \( \pi \).

In our earlier step-by-step solution, we had a total circle area of \( 72 \, \mathrm{cm}^2 \). If you know the total area and wish to find the radius, you rearrange the formula to solve for \( r \). This calculation is crucial for the next steps in solving problems involving sectors.

Central Angle in Radians

Understanding central angles is essential when working with sectors of a circle. A circle’s complete rotation measures \( 2\pi \) radians. A radian is a way to describe angles, much like degrees. It's based on the radius of the circle. In sector calculations, the central angle determines what fraction of the circle's area the sector covers.

In our exercise, the central angle was given as \( \theta = \frac{\pi}{6} \). This means the angle is one twelfth of a full circle, as \( \frac{\pi}{6} \) radians equals \( 30 \) degrees. To calculate the area of a sector, this angle is used to figure out what fraction of the full circle’s area is occupied by the sector. The larger the angle, the larger the sector, and vice versa.

Radius Calculation

Calculating the radius of a circle is often an intermediate step when finding sector areas. When you have the area of the circle, like \( 72 \, \mathrm{cm}^2 \) in this case, and you need the radius, you use the circle area formula in reverse.

• From \( \pi r^2 = 72 \), solve for \( r^2 \): \( r^2 = \frac{72}{\pi} \).
• Then, find \( r \) by taking the square root of \( \frac{72}{\pi} \).

With \( r \) calculated, you can use it in other formulas, like those for sector areas or the circumference. In our exercise, determining \( r \) was essential for substituting into the sector area formula and finding the answer. Knowing how to manipulate these fundamental formulas is key to mastering circle-related problems.