Arc CD Is $\frac{2}{3}$ Of The Circumference Of A Circle. What Is The Radian Measure Of The Central Angle?A. $\frac{2 \pi}{3}$ Radians B. $\frac{3 \pi}{4}$ Radians C. $\frac{4 \pi}{3}$ Radians D. $\frac{3

Introduction

In geometry, the relationship between the arc length and the central angle of a circle is a fundamental concept. The arc length is a measure of the distance along the circumference of a circle, while the central angle is the angle formed by two radii that intersect at the center of the circle. In this article, we will explore the relationship between the arc length and the central angle, and use this relationship to find the radian measure of the central angle.

The Formula for Arc Length

The formula for arc length is given by:

$$L = \frac{\theta}{360} \times 2\pi r$$

where $L$ is the arc length, $\theta$ is the central angle in degrees, $r$ is the radius of the circle, and $\pi$ is a mathematical constant approximately equal to 3.14.

The Relationship Between Arc Length and Central Angle

We are given that the arc length is $\frac{2}{3}$ of the circumference of the circle. The circumference of a circle is given by:

$$C = 2\pi r$$

where $C$ is the circumference and $r$ is the radius of the circle.

Since the arc length is $\frac{2}{3}$ of the circumference, we can write:

$$L = \frac{2}{3} \times 2\pi r$$

Substituting the formula for arc length, we get:

$$\frac{\theta}{360} \times 2\pi r = \frac{2}{3} \times 2\pi r$$

Simplifying the equation, we get:

$$\frac{\theta}{360} = \frac{2}{3}$$

Finding the Central Angle in Degrees

To find the central angle in degrees, we can multiply both sides of the equation by 360:

$$\theta = \frac{2}{3} \times 360$$

Simplifying the equation, we get:

$$\theta = 240$$

Converting the Central Angle to Radians

To convert the central angle from degrees to radians, we can use the following formula:

$$\theta_{\text{radians}} = \frac{\pi}{180} \times \theta_{\text{degrees}}$$

Substituting the value of $\theta$ in degrees, we get:

$$\theta_{\text{radians}} = \frac{\pi}{180} \times 240$$

Simplifying the equation, we get:

$$\theta_{\text{radians}} = \frac{4\pi}{3}$$

Conclusion

In this article, we explored the relationship between the arc length and the central angle of a circle. We used this relationship to find the radian measure of the central angle, which is $\frac{4\pi}{3}$ radians.

Answer

The correct answer is:

• C. $\frac{4 \pi}{3}$ radians

Discussion

This problem requires a good understanding of the relationship between arc length and central angle. The key concept is that the arc length is proportional to the central angle, and the proportionality constant is the ratio of the arc length to the circumference of the circle.

In this problem, we were given that the arc length is $\frac{2}{3}$ of the circumference of the circle. We used this information to find the central angle in degrees, and then converted it to radians.

This problem is a good example of how to use the relationship between arc length and central angle to solve problems in geometry. It requires a good understanding of the formulas and concepts involved, as well as the ability to apply them to solve problems.

Additional Resources

For more information on the relationship between arc length and central angle, see the following resources:

• Wikipedia: Arc length
• Math Open Reference: Arc length
• Khan Academy: Arc length and central angle

Related Problems

• Problem: Find the arc length of a circle with a central angle of 60 degrees
• Problem: Find the central angle of a circle with an arc length of 10 cm
• Problem: Find the circumference of a circle with a radius of 5 cm
  Arc CD is $\frac{2}{3}$ of the circumference of a circle. What is the radian measure of the central angle? Q&A =====================================================================================

Q: What is the relationship between arc length and central angle?

A: The arc length is proportional to the central angle, and the proportionality constant is the ratio of the arc length to the circumference of the circle.

Q: How do you find the central angle in degrees?

A: To find the central angle in degrees, you can use the formula:

$$\theta = \frac{L}{C} \times 360$$

where $L$ is the arc length, $C$ is the circumference of the circle, and $\theta$ is the central angle in degrees.

Q: How do you convert the central angle from degrees to radians?

A: To convert the central angle from degrees to radians, you can use the formula:

$$\theta_{\text{radians}} = \frac{\pi}{180} \times \theta_{\text{degrees}}$$

Q: What is the radian measure of the central angle if the arc length is $\frac{2}{3}$ of the circumference of the circle?

A: To find the radian measure of the central angle, you can use the formula:

$$\theta_{\text{radians}} = \frac{2\pi}{3}$$

Q: Why is the radian measure of the central angle $\frac{4\pi}{3}$ radians?

A: The radian measure of the central angle is $\frac{4\pi}{3}$ radians because the arc length is $\frac{2}{3}$ of the circumference of the circle, and the proportionality constant is the ratio of the arc length to the circumference of the circle.

Q: What is the relationship between the arc length and the central angle in radians?

A: The arc length is proportional to the central angle in radians, and the proportionality constant is the ratio of the arc length to the circumference of the circle.

Q: How do you find the arc length of a circle with a central angle of 60 degrees?

A: To find the arc length of a circle with a central angle of 60 degrees, you can use the formula:

$$L = \frac{\theta}{360} \times 2\pi r$$

where $L$ is the arc length, $\theta$ is the central angle in degrees, $r$ is the radius of the circle, and $\pi$ is a mathematical constant approximately equal to 3.14.

Q: How do you find the central angle of a circle with an arc length of 10 cm?

A: To find the central angle of a circle with an arc length of 10 cm, you can use the formula:

$$\theta = \frac{L}{r} \times 360$$

where $L$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the central angle in degrees.

Q: What is the circumference of a circle with a radius of 5 cm?

A: The circumference of a circle with a radius of 5 cm is:

$$C = 2\pi r = 2\pi \times 5 = 10\pi$$

Conclusion

In this article, we have discussed the relationship between arc length and central angle, and how to find the radian measure of the central angle. We have also answered some common questions related to this topic.

Additional Resources

For more information on the relationship between arc length and central angle, see the following resources:

• Wikipedia: Arc length
• Math Open Reference: Arc length
• Khan Academy: Arc length and central angle

Related Problems

• Problem: Find the arc length of a circle with a central angle of 60 degrees
• Problem: Find the central angle of a circle with an arc length of 10 cm
• Problem: Find the circumference of a circle with a radius of 5 cm