Calculate Radian Measure Arc Length Is 2/3 Circumference

This article delves into the relationship between arc length, central angles, and radians in circles. We will explore how to calculate the radian measure of a central angle when given the fraction of the circumference represented by the arc. The key concept we'll unravel is the direct proportionality between the arc length and the central angle subtended by it. We will start by defining some fundamental terms, then we will illustrate how to solve the problem. Finally, we will provide a step-by-step solution to the problem "Arc CD is 2/3 of the circumference of a circle. What is the radian measure of the central angle?", offering a detailed explanation to enhance understanding. We will also provide some similar problems with detailed explanations.

Understanding the Basics

Before diving into the problem, let's establish a solid foundation by defining essential terms. A circle is a closed two-dimensional figure formed by all points in a plane that are at a fixed distance from a central point. The circumference of a circle is the distance around the circle. An arc is a portion of the circumference of a circle. A central angle is an angle whose vertex is at the center of the circle. The measure of a central angle is often expressed in degrees or radians. A radian is a unit of angular measure defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. There are 2π radians in a full circle, which corresponds to 360 degrees.

In this context, understanding the relationship between arc length, central angles, and radians is crucial. The arc length (*s*) is directly proportional to the central angle (*θ*) it subtends, provided the angle is measured in radians. This relationship is expressed by the formula:

${ s = rθ }$

where:

• ${ s }$ is the arc length,
• ${ r }$ is the radius of the circle, and
• ${ θ }$ is the central angle in radians.

This formula is the cornerstone of solving problems involving arc lengths and central angles. Let's delve deeper into how this relationship works. The circumference ${ C }$ of a circle is given by the formula ${ C = 2πr }$. If an arc constitutes a fraction of the circumference, the central angle subtended by that arc will be the same fraction of the total angle around the center of the circle, which is ${ 2π }$ radians. For example, if an arc is half the circumference, the central angle will be half of ${ 2π }$, which is ${ π }$ radians. Similarly, if an arc is a quarter of the circumference, the central angle will be a quarter of ${ 2π }$, which is ${ π/2 }$ radians. This direct proportionality makes it easy to calculate the central angle when the arc length is given as a fraction of the circumference.

Solving the Problem: Arc CD and Central Angle

Now, let's tackle the problem: "Arc CD is 2/3 of the circumference of a circle. What is the radian measure of the central angle?". To solve this problem, we need to find the central angle that corresponds to an arc length that is ${ 2/3 }$ of the circle's circumference. We know that the full circumference of a circle corresponds to a central angle of ${ 2π }$ radians. Therefore, if an arc is ${ 2/3 }$ of the circumference, the central angle will be ${ 2/3 }$ of ${ 2π }$ radians.

Here's how we can calculate it:

Central angle = (Fraction of circumference) × (Total radians in a circle)

Central angle = ${ (2/3) × (2π) }$

Central angle = ${ (4π)/3 }$ radians

Therefore, the radian measure of the central angle is ${ (4π)/3 }$ radians. This corresponds to option C in the given choices.

Step-by-Step Solution

To further clarify, let’s break down the solution into a step-by-step process:

1. Identify the given information: The arc CD is ${ 2/3 }$ of the circumference of the circle.

2. Recall the total radians in a circle: A full circle corresponds to ${ 2π }$ radians.

3. Set up the proportion: If the arc is ${ 2/3 }$ of the circumference, the central angle will be ${ 2/3 }$ of the total radians.

4. Calculate the central angle: Multiply the fraction of the circumference by the total radians:

   Central angle = ${ (2/3) × (2π) }$

5. Simplify the expression:

   Central angle = ${ (4π)/3 }$ radians

6. State the answer: The radian measure of the central angle is ${ (4π)/3 }$ radians.

This step-by-step approach ensures a clear and logical solution process. Understanding each step is crucial for mastering the concept and applying it to similar problems.

Similar Problems and Solutions

To reinforce your understanding, let's explore some similar problems and their solutions. These examples will help you apply the concepts we've discussed in different contexts.

Problem 1

An arc is ${ 1/4 }$ of the circumference of a circle. What is the radian measure of the central angle subtended by this arc?

Solution:

1. Identify the given information: The arc is ${ 1/4 }$ of the circumference.

2. Recall the total radians in a circle: ${ 2π }$ radians.

3. Set up the proportion: The central angle will be ${ 1/4 }$ of the total radians.

4. Calculate the central angle:

   Central angle = ${ (1/4) × (2π) }$

5. Simplify the expression:

   Central angle = ${ π/2 }$ radians

Therefore, the radian measure of the central angle is ${ π/2 }$ radians.

Problem 2

An arc is ${ 5/6 }$ of the circumference of a circle. Find the radian measure of the central angle subtended by the arc.

Solution:

1. Identify the given information: The arc is ${ 5/6 }$ of the circumference.

2. Recall the total radians in a circle: ${ 2π }$ radians.

3. Set up the proportion: The central angle will be ${ 5/6 }$ of the total radians.

4. Calculate the central angle:

   Central angle = ${ (5/6) × (2π) }$

5. Simplify the expression:

   Central angle = ${ (5π)/3 }$ radians

Thus, the radian measure of the central angle is ${ (5π)/3 }$ radians.

Problem 3

If the central angle subtended by an arc is ${ (3π)/2 }$ radians, what fraction of the circumference is the arc?

Solution:

1. Identify the given information: The central angle is ${ (3π)/2 }$ radians.

2. Recall the total radians in a circle: ${ 2π }$ radians.

3. Set up the proportion: Fraction of circumference = (Central angle) / (Total radians)

4. Calculate the fraction:

   Fraction = ${ ((3π)/2) / (2π) }$

5. Simplify the expression:

   Fraction = ${ (3π) / (2 × 2π) }$

   Fraction = ${ 3/4 }$

Therefore, the arc is ${ 3/4 }$ of the circumference.

Conclusion

In conclusion, understanding the relationship between arc length, central angles, and radians is fundamental in circle geometry. The formula ${ s = rθ }$ and the concept that a full circle corresponds to ${ 2π }$ radians are essential tools for solving problems involving arcs and central angles. By breaking down problems into step-by-step solutions and practicing with similar examples, you can master these concepts and confidently tackle a wide range of geometric challenges. The problem "Arc CD is 2/3 of the circumference of a circle. What is the radian measure of the central angle?" serves as an excellent example of how to apply these principles, and the additional problems provided offer further opportunities for practice and reinforcement. Remember, the key to success in mathematics is a strong foundation in the basics and consistent practice.