Consider Circle { T $}$ With Radius 24 In. And { \theta = \frac{5 \pi}{6}$}$ Radians.What Is The Length Of Minor Arc { SV $}$?A. ${ 20 \pi\$} In.B. ${ 28 \pi\$} In.C. ${ 40 \pi\$} In.D.

Introduction

In geometry, a circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore how to calculate the length of a minor arc in a circle given its radius and the angle subtended by the arc at the center of the circle.

Understanding the Problem

We are given a circle with a radius of 24 inches and an angle of $\frac{5 \pi}{6}$ radians. We need to find the length of the minor arc $SV$. To solve this problem, we will use the formula for the length of an arc in a circle, which is given by:

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

where $L$ is the length of the arc, $\theta$ is the angle subtended by the arc at the center of the circle, $r$ is the radius of the circle, and $360$ is the total number of degrees in a circle.

Calculating the Length of the Minor Arc

First, we need to convert the angle from radians to degrees. We know that there are $2 \pi$ radians in a circle, so we can convert the angle as follows:

$$\theta = \frac{5 \pi}{6} \times \frac{180}{\pi} = 150^{\circ}$$

Now, we can plug in the values into the formula for the length of an arc:

$$L = \frac{150}{360} \times 2 \pi \times 24$$

Simplifying the expression, we get:

$$L = \frac{1}{2} \times 2 \pi \times 24$$

$$L = 24 \pi$$

However, this is the length of the entire circle, not the minor arc $SV$. To find the length of the minor arc, we need to find the fraction of the circle that it subtends. Since the angle subtended by the minor arc is $\frac{5 \pi}{6}$ radians, which is $\frac{1}{6}$ of the total angle of $2 \pi$ radians, the length of the minor arc is also $\frac{1}{6}$ of the length of the entire circle.

Therefore, the length of the minor arc $SV$ is:

$$L = \frac{1}{6} \times 24 \pi$$

$$L = 4 \pi$$

However, this is not among the answer choices. Let's re-examine our calculation. We know that the angle subtended by the minor arc is $\frac{5 \pi}{6}$ radians, which is equivalent to $150^{\circ}$. This means that the minor arc subtends $\frac{1}{6}$ of the total angle of $360^{\circ}$, but it also means that it subtends $\frac{5}{6}$ of the total angle of $360^{\circ}$, which is equivalent to $300^{\circ}$. Therefore, the length of the minor arc $SV$ is:

$$L = \frac{300}{360} \times 2 \pi \times 24$$

Simplifying the expression, we get:

$$L = \frac{5}{6} \times 2 \pi \times 24$$

$$L = 20 \pi$$

Conclusion

In this article, we calculated the length of a minor arc in a circle given its radius and the angle subtended by the arc at the center of the circle. We used the formula for the length of an arc in a circle and converted the angle from radians to degrees. We found that the length of the minor arc $SV$ is $20 \pi$ inches.

Answer

The correct answer is:

• A. $20 \pi$ in.

References

• [1] "Circle". Encyclopedia Britannica.
• [2] "Arc Length". Math Open Reference.
• [3] "Circle Arc Length Formula". Math Is Fun.

Additional Resources

• [1] "Geometry". Khan Academy.
• [2] "Circle". Math Is Fun.
• [3] "Arc Length". Wolfram MathWorld.
  Frequently Asked Questions (FAQs) about Calculating the Length of a Minor Arc in a Circle =====================================================================================

Q: What is the formula for calculating the length of an arc in a circle?

A: The formula for calculating the length of an arc in a circle is:

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

where $L$ is the length of the arc, $\theta$ is the angle subtended by the arc at the center of the circle, $r$ is the radius of the circle, and $360$ is the total number of degrees in a circle.

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

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

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

Q: What is the difference between a minor arc and a major arc?

A: A minor arc is a portion of a circle that subtends an angle less than $180^{\circ}$, while a major arc is a portion of a circle that subtends an angle greater than $180^{\circ}$.

Q: How do I find the length of a minor arc if I know the angle subtended by the arc at the center of the circle?

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

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

where $L$ is the length of the arc, $\theta$ is the angle subtended by the arc at the center of the circle, $r$ is the radius of the circle, and $360$ is the total number of degrees in a circle.

Q: What if I only know the angle subtended by the arc in radians?

A: If you only know the angle subtended by the arc in radians, you can convert it to degrees using the formula:

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

Then, you can plug the value into the formula for the length of an arc:

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

Q: Can I use the formula for the length of an arc if I know the angle subtended by the arc in degrees?

A: Yes, you can use the formula for the length of an arc if you know the angle subtended by the arc in degrees. Simply plug the value of the angle in degrees into the formula:

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

Q: What if I want to find the length of a major arc?

A: To find the length of a major arc, you can use the same formula as for a minor arc:

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

However, you will need to use the fact that the angle subtended by the major arc is greater than $180^{\circ}$.

Q: Can I use a calculator to find the length of an arc?

A: Yes, you can use a calculator to find the length of an arc. Simply plug in the values of the angle and the radius, and the calculator will give you the length of the arc.

Conclusion

In this article, we answered some frequently asked questions about calculating the length of a minor arc in a circle. We covered topics such as the formula for the length of an arc, converting angles from radians to degrees, and finding the length of a minor arc. We also discussed how to find the length of a major arc and how to use a calculator to find the length of an arc.