Problem 30 - Calculus: One and Several Variables

Question

Show that in a circular disk of radius \(r,\) a sector with a central angle of \(\theta\) radians has area \(\frac{1}{2} r^{2} \theta .\) Take \(\theta\) between 0 and \(2 \pi\) HINT: The ares of the circle is \(\pi r^{2}\).

Step-by-Step Solution

Answer

The area of a sector with a central angle of \(theta\) radians in a disc of radius \(r\) can be determined using the formula \(\frac{1}{2} r^{2} \theta\).

Step 1: Determine the Ratio Between Central Angle and Full Circle

The full circle represents a central angle of \(2\pi\) radians. Thus, the ratio, in terms of the angle, between the sector and the full circle is \(\theta/(2\pi)\).

Step 2: Find the Area of the Full Circle

Given, the area of a full circle is \(\pi r^{2}\). This will be useful in the next step where we calculate the area of the sector.

Step 3: Derive the Formula for the Sector's Area

The area of any sector with a central angle of \(\theta\) radians can be determined by multiplying the area of the full circle with the ratio of the sector's angle to a full circle in radians. Therefore, the formula for the sector's area is \(\frac{\theta}{2\pi} \times \(\pi r^{2}\). This formula simplifies to \(\frac{1}{2} r^{2} \theta\).

Key Concepts

Circle Geometry

Circle geometry involves studying the properties and relations of points, lines, and figures in and around circles. The key elements of a circle include:

• The center, which is the fixed point equidistant from all points on the circle.
• The radius, which is the distance from the center to any point on the circle.
• The diameter, which is twice the radius and the distance across the circle through the center.
• The circumference, which is the total distance around the circle, calculated as \( 2\pi r \).

These elements play a crucial role in understanding various concepts associated with circles, such as sectors and arcs. Sectors are slices of the circle, defined by a central angle and the two radii joining the circle's center to the arc endpoints.

Radians

Radians are a way to measure angles, serving as an alternative to degrees. They provide a natural and seamless connection between linear and angular measurements. Unlike degrees, which divide a circle into 360 parts, radians utilize the circle's radius.The full circle is equivalent to \( 2\pi \) radians. This arises from the relationship of the circumference, \( 2\pi r \), to the radius, \( r \). In simpler terms, if the radius was laid along the circumference, it would take \( 2\pi \) radii to complete the loop.Understanding radians is vital in various mathematical contexts, especially in trigonometry and calculus. They offer a more efficient way to relate angles with the lengths of arcs, as seen in formulas like the area of a sector.

Central Angle

A central angle is an angle whose vertex is located at the center of a circle. This angle spans an arc or a sector and can be measured using degrees or radians.The central angle plays a pivotal role in circle geometry as it directly influences both the length of the arc and the area of the sector. In a complete circle, the central angle is \( 2\pi \) radians or 360 degrees.For a sector, the central angle determines the fraction of the circle's total area or circumference included in the sector. If the central angle is \( \theta \) radians, the sector encompasses \( \theta/2\pi \) of the circle's area. This relationship is fundamental in deriving formulas such as the area of a sector.

Sector Formula

The sector formula provides a method to calculate the area of a sector within a circle. This formula is directly derived from the relationship between the sector's central angle and the full circle.The area of a circle is \( \pi r^2 \). A sector, being a portion of this circle, will have an area proportional to its central angle, \( \theta \). The fraction of this angle compared to the whole circle, \( 2\pi \) radians, is \( \theta/2\pi \).Thus, the area of the sector is the circle's area multiplied by this fraction: \[\text{Area of the sector} = \pi r^2 \times \frac{\theta}{2\pi} = \frac{1}{2} r^2 \theta\]This formula beautifully ties together the concepts of radian measure, the circle's geometry, and algebraic manipulation to simplify and solve problems involving circular sectors.