Understanding the Central Angle and Arc Length

Oct 13, 2025 by magicalceylontours.com 5 views

Introduction

When dealing with circles, it's essential to understand the relationships between various components such as the central angle, arc length, and radius. In this article, we will explore how to find the length of the radius of a circle given a central angle and an arc length. We will use the given information to derive the correct equation and solve for the radius.

Understanding the Central Angle and Arc Length

The central angle is the angle formed by two radii that intersect at the center of the circle. In this case, the central angle is given as 6 radians. The arc length, on the other hand, is the length of the curved line that makes up the arc. The arc length is given as 14 inches.

The Relationship Between Central Angle, Arc Length, and Radius

To find the length of the radius, we need to use the formula that relates the central angle, arc length, and radius. The formula is based on the fact that the arc length is proportional to the central angle and the radius. Mathematically, this can be expressed as:

Arc Length = Central Angle x Radius

Using the given information, we can plug in the values to get:

14 = 6 x r

Solving for the Radius

To solve for the radius, we need to isolate the variable r. We can do this by dividing both sides of the equation by 6:

r = 14 / 6

Conclusion

In conclusion, the correct equation to find the length of the radius of the circle is:

r = 14 / 6

This equation is based on the relationship between the central angle, arc length, and radius. By using this equation, we can find the length of the radius of the circle given a central angle and an arc length.

Discussion

The discussion category for this article is mathematics. The article deals with the concept of circles and the relationships between various components such as the central angle, arc length, and radius. The article provides a step-by-step solution to find the length of the radius of the circle given a central angle and an arc length.

Final Answer

The final answer is B. r = 14 / 6.

Additional Information

The central angle is measured in radians.
The arc length is measured in inches.
The radius is measured in inches.
The formula Arc Length = Central Angle x Radius is used to find the length of the radius.

References

[1] "Circles" by Math Open Reference. Retrieved 2023-12-15.
[2] "Central Angle" by Math Is Fun. Retrieved 2023-12-15.
[3] "Arc Length" by Khan Academy. Retrieved 2023-12-15.
[4] "Radius" by Wolfram MathWorld. Retrieved 2023-12-15.

Introduction

In our previous article, we explored how to find the length of the radius of a circle given a central angle and an arc length. We used the formula Arc Length = Central Angle x Radius to derive the correct equation and solve for the radius. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the central angle, and how is it related to the arc length?

A1: The central angle is the angle formed by two radii that intersect at the center of the circle. It is measured in radians. The arc length is the length of the curved line that makes up the arc. The central angle and arc length are related by the formula Arc Length = Central Angle x Radius.

Q2: How do I convert the central angle from radians to degrees?

A2: To convert the central angle from radians to degrees, you can use the following formula:

Degrees = Radians x (180 / π)

Q3: What is the relationship between the arc length and the radius?

A3: The arc length is directly proportional to the radius. The formula Arc Length = Central Angle x Radius shows that the arc length is equal to the central angle multiplied by the radius.

Q4: Can I use the formula Arc Length = Central Angle x Radius to find the radius if I know the arc length and central angle?

A4: Yes, you can use the formula Arc Length = Central Angle x Radius to find the radius if you know the arc length and central angle. Simply rearrange the formula to solve for the radius:

r = Arc Length / Central Angle

Q5: What is the unit of measurement for the radius?

A5: The unit of measurement for the radius is the same as the unit of measurement for the arc length. In this case, the radius is measured in inches.

Q6: Can I use the formula Arc Length = Central Angle x Radius to find the arc length if I know the radius and central angle?

A6: Yes, you can use the formula Arc Length = Central Angle x Radius to find the arc length if you know the radius and central angle. Simply multiply the radius by the central angle:

Arc Length = Radius x Central Angle

Q7: What is the relationship between the central angle and the arc length?

A7: The central angle and arc length are directly proportional. The formula Arc Length = Central Angle x Radius shows that the arc length is equal to the central angle multiplied by the radius.

Q8: Can I use the formula Arc Length = Central Angle x Radius to find the central angle if I know the arc length and radius?

A8: Yes, you can use the formula Arc Length = Central Angle x Radius to find the central angle if you know the arc length and radius. Simply rearrange the formula to solve for the central angle:

Central Angle = Arc Length / Radius