Problem 40 - Algebra 2 and Trigonometry

Question

The central angle of circle \(O\) has a measure of 4.2 radians and it intercepts an arc whose length is 6.3 meters. What is the length, in meters, of the radius of the circle?

Step-by-Step Solution

Answer

The radius of the circle is 1.5 meters.

Step 1: Understanding the Problem

We are given a circle with a central angle measured in radians and the arc it intercepts. We are to find the radius of the circle.

Step 2: Formula for Arc Length

The formula for the length of an arc (83) in a circle is given by \( L = r \theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians.

Step 3: Substituting Given Values

We know the arc length \( L = 6.3 \) meters and the central angle \( \theta = 4.2 \) radians. Substitute these into the formula: \( 6.3 = r \times 4.2 \).

Step 4: Solving for the Radius

To find \( r \), divide both sides of the equation by the central angle \( \theta \). So, \( r = \frac{6.3}{4.2} \).

Step 5: Performing the Calculation

Divide \( 6.3 \) by \( 4.2 \): \( r = 1.5 \) meters.

Key Concepts

Arc Length

An arc of a circle is simply a portion of its circumference. When discussing arc length, it's important to understand how it relates to both the radius and the central angle of the circle.

• Arc length represents the distance along the curved line forming the arc.
• It is calculated using the formula: \( L = r \theta \), where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.
• This formula shows a direct relationship: the bigger the angle or the larger the radius, the longer the arc.

For example, if you have an arc length derived from a central angle of 4.2 radians and a specific radius, you can quickly calculate how long that segment of the circle's perimeter is. Understanding this will also help you explore properties of circles beyond basic geometry.

Central Angle

The central angle of a circle is pivotal when discussing circle geometry. It is the angle that originates from the center of the circle and is created by two radii.

• Central angle is measured in radians, which is the standard measure used in conjunction with arc length.
• Radian measures provide a proportional relationship between the angle size and arc length.
• For instance, in a full circle (360 degrees), there are \(2\pi\) radians in total.

Understanding radian measures can be quite beneficial since the formula for arc length hinges on this metric. For instance, in the current exercise, knowing the central angle helped us determine the corresponding radius by employing the arc length equation.

Radius of a Circle

The radius is a fundamental component of any circle, serving as the straight-line distance from the circle's center to any point along its edge.

• It is half the diameter and a key element in many geometric formulas, including those used to calculate arc length and area.
• When solving for the radius using arc length and central angles, you apply the formula \( r = \frac{L}{\theta} \), which isolates the radius.
• Determining the radius requires knowing either the arc length or the circle's circumference alongside the measure of the central angle.

In summary, when you know the arc length and the central angle, as demonstrated in the exercise, you can find the radius quickly. Through this task, it becomes clearer how these three elements — arc, angle, and radius — interlink within the geometry of a circle.