Understanding Central Angles and Arc Length

Oct 13, 2025 by talynt.org 60 views

Introduction

When dealing with circles and central angles, it's essential to understand the relationship between the angle and the arc length it intersects. In this article, we will explore how to calculate the arc length intersected by a central angle of $$\frac{\pi}{2}$$ radians in a circle with a radius of 4.5 cm.

Understanding Central Angles and Arc Length

A central angle is an angle formed by two radii that intersect at the center of a circle. The arc length is the length of the arc that the central angle intersects. To calculate the arc length, we need to use the formula:

Arc Length = (central angle / 360) × 2πr

where r is the radius of the circle.

Calculating the Arc Length

Given that the central angle is $$\frac{\pi}{2}$$ radians and the radius is 4.5 cm, we can plug these values into the formula:

Arc Length = ($$\frac{\pi}{2}$$ / 360) × 2π(4.5)

First, let's simplify the fraction:

\frac{\pi}{2}$$ / 360 = $$\frac{\pi}{720}

Now, we can substitute this value into the formula:

Arc Length = ($$\frac{\pi}{720}$$) × 2π(4.5)

Using the value of π as 3.14, we can calculate the arc length:

Arc Length = ($$\frac{3.14}{720}$$) × 2(3.14)(4.5)

Arc Length = ($$\frac{3.14}{720}$$) × 28.26

Arc Length = 0.0125 × 28.26

Arc Length ≈ 0.353 cm

However, we are asked to round our answer to the nearest tenth. Therefore, the arc length is approximately 0.4 cm.

Conclusion

In this article, we calculated the arc length intersected by a central angle of $$\frac{\pi}{2}$$ radians in a circle with a radius of 4.5 cm. We used the formula Arc Length = (central angle / 360) × 2πr and substituted the given values to find the arc length. Our final answer is approximately 0.4 cm.

Discussion

The options provided are A. 0.3 cm, B. 0.7 cm, C. 2.9 cm, and D. 7.1 cm. Our calculated answer is approximately 0.4 cm, which is closest to option A. 0.3 cm. However, it's essential to note that our calculated answer is not exactly 0.3 cm, but rather approximately 0.4 cm.

Final Answer

The final answer is approximately 0.4 cm, which is closest to option A. 0.3 cm.

Introduction

In our previous article, we explored how to calculate the arc length intersected by a central angle of $$\frac{\pi}{2}$$ radians in a circle with a radius of 4.5 cm. In this article, we will answer some frequently asked questions (FAQs) about central angles and arc length.

Q: What is a central angle?

A: A central angle is an angle formed by two radii that intersect at the center of a circle.

Q: How is the arc length related to the central angle?

A: The arc length is the length of the arc that the central angle intersects. To calculate the arc length, we use the formula: Arc Length = (central angle / 360) × 2πr, where r is the radius of the circle.

Q: What is the formula for calculating the arc length?

A: The formula for calculating the arc length is: Arc Length = (central angle / 360) × 2πr.

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

A: To convert a central angle from radians to degrees, you can use the following formula: degrees = radians × (180 / π).

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

A: The central angle and the arc length are directly proportional. As the central angle increases, the arc length also increases.

Q: Can I use the formula for arc length with a central angle in degrees?

A: Yes, you can use the formula for arc length with a central angle in degrees. However, you need to convert the central angle from degrees to radians first.

Q: What is the unit of measurement for arc length?

A: The unit of measurement for arc length is the same as the unit of measurement for the radius of the circle.

Q: Can I use the formula for arc length with a central angle in radians and a radius in inches?

A: Yes, you can use the formula for arc length with a central angle in radians and a radius in inches. However, you need to make sure that the units of measurement are consistent.

Q: What is the significance of the value π in the formula for arc length?

A: The value π is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. It is approximately equal to 3.14.

Q: Can I use a calculator to calculate the arc length?

A: Yes, you can use a calculator to calculate the arc length. However, you need to make sure that the calculator is set to the correct mode (e.g., degrees or radians).

Q: What is the relationship between the central angle and the circumference of the circle?

A: The central angle and the circumference of the circle are related by the formula: arc length = (central angle / 360) × circumference.

Q: Can I use the formula for arc length with a central angle in degrees and a circumference in meters?

A: Yes, you can use the formula for arc length with a central angle in degrees and a circumference in meters. However, you need to make sure that the units of measurement are consistent.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about central angles and arc length. We hope that this article has provided you with a better understanding of the relationship between central angles and arc length.

Final Answer

The final answer is that the arc length is approximately 0.4 cm, which is closest to option A. 0.3 cm.