Converting the Angle from Radians to Degrees

Oct 13, 2025 by magicalceylontours.com 121 views

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. A circle can be divided into two main parts: the major arc and the minor arc. The major arc is the longer arc that is greater than 180 degrees, while the minor arc is the shorter arc that is less than 180 degrees. In this article, we will discuss how to calculate the length of a minor arc in a circle.

Understanding the Problem

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

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

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

Converting the Angle from Radians to Degrees

Before we can use the formula, we need to convert the angle from radians to degrees. We can do this by multiplying the angle in radians by $\frac{180}{\pi}$ .

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

Calculating the Length of the Minor Arc

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

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

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

L = 24 \pi

However, this is the length of the entire circle, not just the minor arc. To find the length of the minor arc, we need to divide the length of the entire circle by 2.

L = \frac{24 \pi}{2}

L = 12 \pi

But this is still not the correct answer. We need to consider that the angle $\theta$ is $\frac{5 \pi}{6}$ radians, which is greater than $\frac{\pi}{2}$ radians. This means that the minor arc SV is actually $\frac{1}{6}$ of the entire circle, not $\frac{1}{2}$ .

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

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

L = 20 \pi

Conclusion

In conclusion, the length of the minor arc SV is $20 \pi$ inches.

References

[1] "Geometry" by Michael Spivak
[2] "Calculus" by Michael Spivak

Additional Resources

[1] Khan Academy: Geometry
[2] MIT OpenCourseWare: Calculus

Frequently Asked Questions

Q: What is the formula for the length of an arc? A: The formula for the length of an arc is given by $L = \frac{\theta}{360} \times 2 \pi r$ .
Q: How do I convert an angle from radians to degrees? A: To convert an angle from radians to degrees, multiply the angle in radians by $\frac{180}{\pi}$ .
Q: How do I calculate the length of a minor arc? A: To calculate the length of a minor arc, use the formula $L = \frac{\theta}{360} \times 2 \pi r$ , where $\theta$ is the angle in degrees and $r$ is the radius of the circle.
Q&A: Calculating the Length of a Minor Arc in a Circle =====================================================

Introduction

In our previous article, we discussed how to calculate the length of a minor arc in a circle. We used the formula $L = \frac{\theta}{360} \times 2 \pi r$ to find the length of the minor arc SV. In this article, we will answer some frequently asked questions about calculating the length of a minor arc.

Q: What is the formula for the length of an arc?

A: The formula for the length of an arc is given by $L = \frac{\theta}{360} \times 2 \pi r$ , where $L$ is the length of the arc, $\theta$ is the angle 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, multiply the angle in radians by $\frac{180}{\pi}$ . For example, if you have an angle of $\frac{5 \pi}{6}$ radians, you can convert it to degrees by multiplying it by $\frac{180}{\pi}$ .

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

A: To calculate the length of a minor arc, use the formula $L = \frac{\theta}{360} \times 2 \pi r$ , where $\theta$ is the angle in degrees and $r$ is the radius of the circle. However, if the angle is greater than $\frac{\pi}{2}$ radians, you need to consider that the minor arc is actually $\frac{1}{6}$ of the entire circle.

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

A: A major arc is the longer arc that is greater than 180 degrees, while a minor arc is the shorter arc that is less than 180 degrees.

Q: How do I know if I have a major arc or a minor arc?

A: To determine if you have a major arc or a minor arc, look at the angle in degrees. If the angle is greater than 180 degrees, you have a major arc. If the angle is less than 180 degrees, you have a minor arc.

Q: Can I use the formula for the length of an arc for any type of circle?

A: Yes, you can use the formula for the length of an arc for any type of circle, including circles with a radius of 0.

Q: What is the length of the entire circle?

A: The length of the entire circle is given by $C = 2 \pi r$ , where $C$ is the circumference of the circle and $r$ is the radius of the circle.

Q: How do I find the length of a sector of a circle?

A: To find the length of a sector of a circle, use the formula $L = \frac{\theta}{360} \times 2 \pi r$ , where $\theta$ is the angle in degrees and $r$ is the radius of the circle.

Conclusion

In conclusion, calculating the length of a minor arc in a circle can be a complex task, but with the right formula and a little practice, you can become a pro at it. Remember to always convert the angle from radians to degrees and to consider that the minor arc is actually $\frac{1}{6}$ of the entire circle if the angle is greater than $\frac{\pi}{2}$ radians.

References

[1] "Geometry" by Michael Spivak
[2] "Calculus" by Michael Spivak

Additional Resources

[1] Khan Academy: Geometry
[2] MIT OpenCourseWare: Calculus

Frequently Asked Questions

Q: What is the formula for the length of an arc? A: The formula for the length of an arc is given by $L = \frac{\theta}{360} \times 2 \pi r$ .
Q: How do I convert an angle from radians to degrees? A: To convert an angle from radians to degrees, multiply the angle in radians by $\frac{180}{\pi}$ .
Q: How do I calculate the length of a minor arc? A: To calculate the length of a minor arc, use the formula $L = \frac{\theta}{360} \times 2 \pi r$ , where $\theta$ is the angle in degrees and $r$ is the radius of the circle.