Problem 65 - Algebra and Trigonometry

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: Understand the Relationship Between Sector and Circle Area

The area of a sector (A_sector) is a fraction of the circle's total area (A_circle). This fraction is based on the ratio of the sector's central angle, \(\theta\), to the total angle of a circle, which is \(2\pi\) radians. Thus, the formula for the area of a sector is \(A_{\text{sector}} = \frac{\theta}{2\pi} \times A_{\text{circle}}\).

Step 2: Identify Given Values

We know the total area of the circle is \(A_{\text{circle}} = 72\, \mathrm{cm}^2\) and the central angle \(\theta = \frac{\pi}{6}\) radians.

Step 3: Substitute Values into the Formula

Substitute the known values into the sector area formula: \(A_{\text{sector}} = \frac{\pi/6}{2\pi} \times 72\, \mathrm{cm}^2\).

Step 4: Simplify the Expression

First, simplify the fraction: \(\frac{\pi/6}{2\pi} = \frac{1}{12}\). Then, calculate the area of the sector: \(A_{\text{sector}} = \frac{1}{12} \times 72\, \mathrm{cm}^2\).

Step 5: Calculate the Final Answer

Finally, perform the multiplication: \(A_{\text{sector}} = \frac{1 \times 72}{12} = 6\, \mathrm{cm}^2\).

Key Concepts

Circle Area

The area of a circle is a fundamental geometric concept. It measures the total space contained within the circle's boundary. To determine the circle's area, we usually use the formula: \[ A = \pi r^2 \]where \( r \) is the radius of the circle. This formula shows the proportional relationship between the area and the square of the radius.

• \( \pi \) (Pi) is a constant approximately equal to 3.14159.
• The circle's radius is the distance from the center to any point on the perimeter.

It’s important to remember that all points on the circle are equidistant from the center, a unique property essential to its symmetry. Knowing how to calculate the area of a circle is crucial for solving related problems, such as finding the area of a sector.

Central Angle

In the context of a circle, a central angle is the angle formed between two radii that extend from the center of the circle to its circumference. This angle helps specify what portion of the circle you're dealing with.

• The total central angle of a full circle is \( 360^\circ \) or \( 2\pi \) radians.
• A sector is a pie-shaped part of the circle, defined by the area enclosed between two radii and the circle's circumference.

For instance, if the central angle of a sector is \( \frac{\pi}{6} \), this means it constitutes \(\frac{1}{12}\) of the circle.Understanding central angles is essential for calculating the areas of various sections in circular geometries.

Radian Measure

Radians are a unit of measurement for angles used primarily in trigonometry and circle mathematics. Unlike degrees, radians link directly to the circle's radius, making calculations involving arcs and sectors intuitive and straightforward.

• A full circle is \( 2\pi \) radians. Thus, dividing by radians provides proportions helpful in geometry.
• One radian is the angle formed when the arc's length equals the radius.

Using radians simplifies angular calculations and transformations. This approach is particularly useful when determining the area of a sector, where the central angle in radians provides a direct fraction of the total circle. Keep in mind that in the conversion process, radians provide a more direct and natural measure relative to the circle's geometry.