Problem 59 - Algebra and Trigonometry

Question

Find the radius of the circle if an arc of length 6 \(\mathrm{m}\) on the circle subtends a central angle of \(\pi / 6 \mathrm{rad}\) .

Step-by-Step Solution

Answer

The radius is \( \frac{36}{\pi} \) meters.

Step 1: Understand the relationship

In a circle, the length of an arc \( L \) is related to the radius \( r \) and the subtended angle \( \theta \) (in radians) by the formula \( L = r \theta \). Our goal is to find the radius \( r \).

Step 2: Substitute known values

We know the arc length \( L \) is 6 meters and the angle \( \theta \) is \( \pi/6 \) radians. Substitute these into the formula: \( 6 = r \cdot (\pi/6) \).

Step 3: Solve for the radius

Rearrange the equation to solve for \( r \): \( r = \frac{6}{\pi/6} \).

Step 4: Simplify

Simplify the calculation: \( r = \frac{6 \times 6}{\pi} = \frac{36}{\pi} \) meters.

Key Concepts

Arc Length

The arc length is a portion of the circumference of a circle, much like a slice of pie from a pie chart. It is determined by two main factors: the size of the central angle and the circle's radius. The formula to calculate arc length \( L \) is:

• \( L = r \theta \)

where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians. If the angle is large, the arc will be longer, and if the radius is large, then also expect a longer arc. Therefore, understanding this relationship helps in calculating one value if the others are known, making it a fundamental concept in circle geometry.

Central Angle

A central angle is an angle whose vertex is located at the center of the circle. This type of angle effectively "opens" a segment of the circle, framing an arc. The size of this angle directly affects the length of the arc it intercepts.

• The larger the central angle, the longer its corresponding arc.
• Although measured in degrees in everyday use, radians are preferred in mathematical formulas.

This angle is crucial in our calculations since it helps define the size of both the arc and the area of the sector.

Radian Measure

Radians offer a more natural measure for angles compared to degrees, especially in mathematics. The radian measure is defined as the angle made when the radius is wrapped around the circle's circumference.

• One full circle is \( 2\pi \) radians.
• Knowing the equivalence, \( \pi \) radians equal \( 180^\circ \), helps in converting angles.

In this context, we find the formula for arc length heavily relies on using radians for the central angle, making it an indispensable concept.Understanding radians is pivotal for mastering topics in trigonometry, calculus, and other advanced fields where circles play a crucial role.

Circle Geometry

Circle geometry explores the properties and relationships of various parts of a circle. It includes the study of radii, diameters, chords, tangents, sectors, and segments. Fundamental concepts include:

• The radius is a straight line from the center to any point on the circle.
• The diameter is twice the radius and divides the circle into two equal halves.
• The circumference, which can be calculated as \( 2\pi r \).

Each element plays a role in solving geometric problems like the one presented. Understanding these basic components allows us to find unknown elements like the radius, given other measurements such as arc length and central angle.