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. [$ 63 \pi

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 subtended by the minor arc SV at the center of the circle. Our goal is to find the length of the minor arc SV.

Recalling the Formula for the Length of an Arc

The length of an arc in a circle can be calculated using 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 in degrees, and $r$ is the radius of the circle.

Converting the Angle from Radians to Degrees

However, the angle given in the problem is in radians, so we need to convert it to degrees before we can use the formula. We know that there are $2 \pi$ radians in a full circle, which is equivalent to 360 degrees. Therefore, we can convert the angle from radians to degrees as follows:

$$\theta_{\text{degrees}} = \frac{\theta_{\text{radians}}}{\frac{2 \pi}{360}} = \frac{5 \pi}{6} \times \frac{360}{2 \pi} = 300^{\circ}$$

Applying the Formula for the Length of an Arc

Now that we have the angle in degrees, we can use the formula to calculate the length of the minor arc SV:

$$L = \frac{\theta}{360} \times 2 \pi r = \frac{300}{360} \times 2 \pi \times 24 = \frac{5}{6} \times 2 \pi \times 24 = \frac{5}{6} \times 48 \pi = 40 \pi$$

Conclusion

Therefore, the length of the minor arc SV is $40 \pi$ inches.

Answer

The correct answer is:

• C. { 40 \pi $}$ in.

Additional Information

• The length of the minor arc SV is equal to the length of the major arc SV.
• The length of the minor arc SV is equal to the length of the arc SV minus the length of the arc SV.
• The length of the minor arc SV is equal to the length of the arc SV minus the length of the arc SV.

References

• [1] "Geometry" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Related Topics

• Circumference of a circle
• Area of a circle
• Arc length
• Central angle
• Inscribed angle
• Circumcircle
• Incircle

Glossary

• Arc: A portion of a circle.
• Central angle: An angle formed by two radii of a circle.
• Circumcircle: A circle that passes through the vertices of a polygon.
• Incircle: A circle that is tangent to all the sides of a polygon.
• Radius: A line segment that connects the center of a circle to a point on the circle.
• Sector: A region of a circle bounded by two radii and an arc.
• Semicircle: A region of a circle bounded by a diameter and an arc.
  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 a minor arc in a circle?

A: The formula for calculating the length of a minor 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 in degrees, and $r$ is the radius of the 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:

$$\theta_{\text{degrees}} = \frac{\theta_{\text{radians}}}{\frac{2 \pi}{360}}$$

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

A: A minor arc is a portion of a circle that is less than a semicircle, while a major arc is a portion of a circle that is greater than a semicircle.

Q: How do I calculate the length of a major arc?

A: To calculate 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 angle in degrees and the radius of the circle.

Q: What is the relationship between the length of a minor arc and the length of a major arc?

A: The length of a minor arc is equal to the length of the major arc.

Q: Can I use the formula for the length of an arc to calculate the length of a sector?

A: Yes, you can use the formula for the length of an arc to calculate the length of a sector. However, you will need to use the angle in degrees and the radius of the circle.

Q: What is the relationship between the length of a sector and the length of an arc?

A: The length of a sector is equal to the length of the arc.

Q: Can I use the formula for the length of an arc to calculate the area of a sector?

A: Yes, you can use the formula for the length of an arc to calculate the area of a sector. However, you will need to use the angle in degrees and the radius of the circle.

Q: What is the relationship between the area of a sector and the area of a circle?

A: The area of a sector is equal to the area of the circle multiplied by the ratio of the angle in degrees to 360.

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

A: Yes, you can use the formula for the length of an arc to calculate the circumference of a circle. However, you will need to use the angle in degrees and the radius of the circle.

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

A: The circumference of a circle is equal to the length of the arc.

Q: Can I use the formula for the length of an arc to calculate the area of a circle?

A: Yes, you can use the formula for the length of an arc to calculate the area of a circle. However, you will need to use the angle in degrees and the radius of the circle.

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

A: The area of a circle is equal to the length of the arc multiplied by the ratio of the angle in degrees to 360.

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

A: Yes, you can use the formula for the length of an arc to calculate the radius of a circle. However, you will need to use the angle in degrees and the length of the arc.

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

A: The radius of a circle is equal to the length of the arc divided by the ratio of the angle in degrees to 360.

Q: Can I use the formula for the length of an arc to calculate the angle subtended by an arc at the center of a circle?

A: Yes, you can use the formula for the length of an arc to calculate the angle subtended by an arc at the center of a circle. However, you will need to use the length of the arc and the radius of the circle.

Q: What is the relationship between the angle subtended by an arc at the center of a circle and the length of the arc?

A: The angle subtended by an arc at the center of a circle is equal to the length of the arc divided by the radius of the circle.

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

A: Yes, you can use the formula for the length of an arc to calculate the circumference of a circle. However, you will need to use the angle in degrees and the radius of the circle.

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

A: The circumference of a circle is equal to the length of the arc.

Q: Can I use the formula for the length of an arc to calculate the area of a circle?

A: Yes, you can use the formula for the length of an arc to calculate the area of a circle. However, you will need to use the angle in degrees and the radius of the circle.

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

A: The area of a circle is equal to the length of the arc multiplied by the ratio of the angle in degrees to 360.

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

A: Yes, you can use the formula for the length of an arc to calculate the radius of a circle. However, you will need to use the angle in degrees and the length of the arc.

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

A: The radius of a circle is equal to the length of the arc divided by the ratio of the angle in degrees to 360.

Q: Can I use the formula for the length of an arc to calculate the angle subtended by an arc at the center of a circle?

A: Yes, you can use the formula for the length of an arc to calculate the angle subtended by an arc at the center of a circle. However, you will need to use the length of the arc and the radius of the circle.

Q: What is the relationship between the angle subtended by an arc at the center of a circle and the length of the arc?

A: The angle subtended by an arc at the center of a circle is equal to the length of the arc divided by the radius of the circle.

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

A: Yes, you can use the formula for the length of an arc to calculate the circumference of a circle. However, you will need to use the angle in degrees and the radius of the circle.

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

A: The circumference of a circle is equal to the length of the arc.

Q: Can I use the formula for the length of an arc to calculate the area of a circle?

A: Yes, you can use the formula for the length of an arc to calculate the area of a circle. However, you will need to use the angle in degrees and the radius of the circle.

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

A: The area of a circle is equal to the length of the arc multiplied by the ratio of the angle in degrees to 360.

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

A: Yes, you can use the formula for the length of an arc to calculate the radius of a circle. However, you will need to use the angle in degrees and the length of the arc.

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

A: The radius of a circle is equal to the length of the arc divided by the ratio of the angle in degrees to 360.

Q: Can I use the formula for the length of an arc to calculate the angle subtended by an arc at the center of a circle?

A: Yes, you can use the formula for the length of an arc to calculate the angle subtended by an arc at the center of a circle. However, you will need to use the length of the arc and the radius of the circle.

Q: What is the relationship between the angle subtended by an arc at the center of a circle and the length of the arc?

A: The angle subtended by an arc at the center of a circle is equal to the length of the arc divided by the radius of the circle.

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