Problem 64 - Algebra and Trigonometry

Question

A sector of a circle has a central angle of \(60^{\circ} .\) Find the area of the sector if the radius of the circle is \(3 \mathrm{mi} .\)

Step-by-Step Solution

Answer

The area of the sector is \(\frac{3}{2}\pi \text{ mi}^2\).

Step 1: Understand the Circle and Sector

A sector of a circle is a portion of the circle enclosed by two radii and the corresponding arc. The given problem specifies that the central angle of the sector is \(60^{\circ}\) and the radius of the circle, denoted as \(r\), is \(3 \text{ mi}\). To solve for the area of the sector, we need to use the formula for the area of a sector.

Step 2: Recall the Formula for Sector Area

The formula for the area of a sector with a central angle \(\theta\) in degrees is \( A = \frac{\theta}{360} \cdot \pi r^2 \). This formula derives from the fact that the sector is a fraction of the entire circle's area. Here, \(\theta\) is \(60^{\circ}\) and \(r = 3 \text{ mi}\).

Step 3: Substitute the Known Values into the Formula

Substitute \(\theta = 60^{\circ}\) and \(r = 3\) mi into the formula: \[ A = \frac{60}{360} \cdot \pi \cdot (3)^2 \] This calculation will give us the area of the sector.

Step 4: Simplify the Expression

First, simplify the fraction \(\frac{60}{360}\):\[ \frac{60}{360} = \frac{1}{6} \]Next, calculate \( (3)^2 \): \[ (3)^2 = 9 \]Substitute back:\[ A = \frac{1}{6} \cdot \pi \cdot 9 \]

Step 5: Calculate the Area of the Sector

Finish the calculation by multiplying the values:\[ A = \frac{1}{6} \cdot 9 \cdot \pi = \frac{9}{6} \cdot \pi = \frac{3}{2} \cdot \pi \]So, the area of the sector is \(\frac{3}{2}\pi \text{ mi}^2\).

Key Concepts

Understanding the Central Angle

In a circle, the central angle is the angle whose vertex is at the center of the circle and whose sides are radii of the circle. This angle plays a crucial role when working with segments and sectors of a circle. When it comes to calculating areas related to circles, knowing the central angle is key because it helps determine what fraction of the circle you are dealing with. For example, if the central angle is smaller, the sector or segment will occupy a smaller part of the circle, and vice versa. In the original exercise, the central angle is given as \(60^{\circ}\). This means that the sector represents \(\frac{60}{360} = \frac{1}{6}\) of the entire circle.
Understanding this angle helps us figure out how much of the circle's area we need to consider when calculating the sector's area.

Exploring the Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. It is a crucial measurement in geometry because it helps to calculate various attributes of a circle, such as the diameter, circumference, and area. The radius is half of the diameter, which is the longest distance across a circle. You can find the circumference using the formula \(C = 2\pi r\) and the area with \(A = \pi r^2\), where \(r\) is the radius.When calculating the area of a sector, the radius is squared and multiplied by pi to find the full circle's area, then divided according to the sector's central angle. In our problem, the radius is given as \(3\ \text{mi}\). This tells us that if we were dealing with the entire circle, it would have an area of \(\pi (3^2) = 9\pi\ \text{mi}^2\). But since we are focusing on a sector, we have to adjust using the central angle.

The Basics of Circle Geometry

Circle geometry deals with the properties and relations of all parts of a circle, including angles, radii, chords, and sectors. It forms the basis for understanding many geometric shapes and figures that include or are derived from circles. Some key components of circle geometry are:

• Radius: As discussed, it is the distance from center to the edge.
• Diameter: A line segment that passes through the center and touches two points on the boundary.
• Circumference: The perimeter or boundary length of the circle.
• Arc: A continuous portion of the circle's circumference.
• Sector: An area enclosed by two radii and an arc.
• Segment: A region bounded by a chord and an arc lying between the chord's endpoints.

For any geometry problem involving a circle, it's essential to understand these components. The calculations often revolve around determining lengths, areas, or angles based on these properties. In this exercise, understanding the relationships between the radius, central angle, and the segment they form helped in finding the area of the sector accurately.