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. $ 3 Π 4 \frac{3 \pi}{4} 4 3 Π ​ [/tex] Radians C. $\frac{4 \pi}{3}$ Radians

Introduction

In geometry, the relationship between the arc length of a circle and its central angle 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 we will use this relationship to find the radian measure of the central angle.

The Formula for Arc Length

The formula for the arc length of a circle is given by:

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

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

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 is a classic example of how to use the relationship between arc length and central angle to find the radian measure of the central angle. The key concept is to use the formula for arc length and the relationship between arc length and central angle to find the central angle in degrees, and then convert it to radians.

Additional Resources

For more information on the relationship between arc length and central angle, you can refer to the following resources:

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

Related Problems

If you want to practice more problems on the relationship between arc length and central angle, you can try the following problems:

• Find the radian measure of the central angle if the arc length is $\frac{1}{2}$ of the circumference of the circle.
• Find the radian measure of the central angle if the arc length is $\frac{3}{4}$ of the circumference of the circle.
• Find the radian measure of the central angle if the arc length is $\frac{2}{5}$ of the circumference of the circle.
  Q&A: Arc Length and Central Angle =====================================

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

A: The arc length of a circle is directly proportional to the central angle. 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, and $r$ is the radius of the circle.

Q: How do I find the central angle in degrees if I know the arc length and radius?

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

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

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

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

A: To convert the central angle from degrees to radians, you can use the following 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: What is the radian measure of the central angle if the arc length is $\frac{1}{2}$ of the circumference of the circle?

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

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

Q: What is the radian measure of the central angle if the arc length is $\frac{3}{4}$ of the circumference of the circle?

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

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

Q: How do I find the arc length if I know the central angle and radius?

A: To find the arc length, you can use the formula:

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

Q: What is the relationship between the arc length and the circumference of a circle?

A: The arc length is directly proportional to the circumference of a circle. 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, and $r$ is the radius of the circle.

Q: Can I use the formula for arc length to find the circumference of a circle?

A: Yes, you can use the formula for arc length to find the circumference of a circle. If the arc length is equal to the circumference, then the central angle is equal to 360 degrees.

Q: What is the radian measure of the central angle if the arc length is equal to the circumference of the circle?

A: The radian measure of the central angle is equal to $2\pi$ radians.

Q: Can I use the formula for arc length to find the radius of a circle?

A: Yes, you can use the formula for arc length to find the radius of a circle. If you know the arc length and central angle, you can use the formula:

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

to find the radius of the circle.

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

A: The arc length is directly proportional to the central angle in radians. The formula for arc length is given by:

$$L = \theta \times r$$

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

Q: Can I use the formula for arc length to find the central angle in radians?

A: Yes, you can use the formula for arc length to find the central angle in radians. If you know the arc length and radius, you can use the formula:

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

to find the central angle in radians.

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:

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

Q: What is the radian measure of the central angle if the arc length is $\frac{1}{2}$ 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: What is the radian measure of the central angle if the arc length is $\frac{3}{4}$ of the circumference of the circle?

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

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