Arc CD Is { \frac{1}{4}$}$ Of The Circumference Of A Circle. What Is The Radian Measure Of The Central Angle?A. { \frac{\pi}{4}$}$ Radians B. { \frac{\pi}{2}$}$ Radians C. ${ 2\pi\$} Radians D.

Introduction

In geometry, the relationship between arc length and central angles in circles is a fundamental concept. The arc length of a circle is directly proportional to the central angle subtended by the arc. In this article, we will explore the relationship between arc length and central angles, and use this knowledge to solve a problem involving the arc CD of a circle.

The Relationship Between Arc Length and Central Angles

The arc length of a circle is given by the formula:

L = rθ

where L is the arc length, r is the radius of the circle, and θ is the central angle subtended by the arc in radians.

Understanding Radians

Radians are a unit of measurement for angles, and are defined as the ratio of the arc length to the radius of the circle. In other words, if an arc of length L subtends a central angle θ, then the measure of the angle in radians is given by:

θ = L/r

The Problem

The problem states that the arc CD is $\frac{1}{4}$ of the circumference of a circle. We need to find the radian measure of the central angle subtended by the arc CD.

Step 1: Find the Circumference of the Circle

The circumference of a circle is given by the formula:

C = 2πr

where C is the circumference, and r is the radius of the circle.

Step 2: Find the Length of the Arc CD

The problem states that the arc CD is $\frac{1}{4}$ of the circumference of the circle. Therefore, the length of the arc CD is:

L = $\frac{1}{4}$C

Step 3: Find the Radian Measure of the Central Angle

We can now use the formula for the radian measure of the central angle:

θ = L/r

Substituting the values, we get:

θ = ($\frac{1}{4}$C)/r

θ = ($\frac{1}{4}$)(2πr)/r

θ = $\frac{1}{2}$π

θ = $\frac{\pi}{2}$ radians

Therefore, the radian measure of the central angle subtended by the arc CD is $\frac{\pi}{2}$ radians.

Conclusion

In this article, we explored the relationship between arc length and central angles in circles. We used this knowledge to solve a problem involving the arc CD of a circle, and found that the radian measure of the central angle is $\frac{\pi}{2}$ radians.

Key Takeaways

• The arc length of a circle is directly proportional to the central angle subtended by the arc.
• The radian measure of the central angle is given by the formula: θ = L/r.
• The circumference of a circle is given by the formula: C = 2πr.
• The length of an arc is given by the formula: L = $\frac{1}{4}$C.

Frequently Asked Questions

Q: What is the relationship between arc length and central angles in circles? A: The arc length of a circle is directly proportional to the central angle subtended by the arc.

Q: How do you find the radian measure of the central angle? A: You can use the formula: θ = L/r.

Q: What is the circumference of a circle? A: The circumference of a circle is given by the formula: C = 2πr.

Q: What is the length of an arc? A: The length of an arc is given by the formula: L = $\frac{1}{4}$C.

References

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

Glossary

• Arc length: The length of an arc of a circle.
• Central angle: The angle subtended by an arc at the center of a circle.
• Circumference: The distance around a circle.
• Radian: A unit of measurement for angles, defined as the ratio of the arc length to the radius of the circle.
  Arc CD is $\frac{1}{4}$ of the Circumference of a Circle: Q&A ================================================================

Introduction

In our previous article, we explored the relationship between arc length and central angles in circles. We used this knowledge to solve a problem involving the arc CD of a circle, and found that the radian measure of the central angle is $\frac{\pi}{2}$ radians. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the relationship between arc length and central angles in circles?

A: The arc length of a circle is directly proportional to the central angle subtended by the arc. This means that if the central angle is doubled, the arc length will also double.

Q: How do you find the radian measure of the central angle?

A: You can use the formula: θ = L/r, where θ is the central angle in radians, L is the arc length, and r is the radius of the circle.

Q: What is the circumference of a circle?

A: The circumference of a circle is given by the formula: C = 2πr, where C is the circumference, and r is the radius of the circle.

Q: What is the length of an arc?

A: The length of an arc is given by the formula: L = $\frac{1}{4}$C, where L is the arc length, and C is the circumference of the circle.

Q: How do you find the central angle if you know the arc length and radius?

A: You can use the formula: θ = L/r, where θ is the central angle in radians, L is the arc length, and r is the radius of the circle.

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

A: The central angle is directly proportional to the arc length. This means that if the arc length is doubled, the central angle will also double.

Q: Can you give an example of how to use the formula θ = L/r?

A: Yes, let's say we have a circle with a radius of 5 units, and an arc length of 10 units. We can use the formula to find the central angle:

θ = L/r θ = 10/5 θ = 2 radians

Q: What is the radian measure of the central angle if the arc length is $\frac{1}{2}$ of the circumference?

A: If the arc length is $\frac{1}{2}$ of the circumference, then the central angle is $\frac{1}{2}$ of the total angle of the circle, which is 2π radians. Therefore, the radian measure of the central angle is π radians.

Q: Can you give an example of how to use the formula θ = L/r to find the central angle if the arc length is $\frac{1}{2}$ of the circumference?

A: Yes, let's say we have a circle with a radius of 5 units, and an arc length of 10 units, which is $\frac{1}{2}$ of the circumference. We can use the formula to find the central angle:

θ = L/r θ = 10/5 θ = 2 radians

Q: What is the radian measure of the central angle if the arc length is $\frac{1}{4}$ of the circumference?

A: If the arc length is $\frac{1}{4}$ of the circumference, then the central angle is $\frac{1}{4}$ of the total angle of the circle, which is 2π radians. Therefore, the radian measure of the central angle is $\frac{1}{2}$π radians.

Q: Can you give an example of how to use the formula θ = L/r to find the central angle if the arc length is $\frac{1}{4}$ of the circumference?

A: Yes, let's say we have a circle with a radius of 5 units, and an arc length of 5 units, which is $\frac{1}{4}$ of the circumference. We can use the formula to find the central angle:

θ = L/r θ = 5/5 θ = 1 radian

Q: What is the radian measure of the central angle if the arc length is $\frac{1}{8}$ of the circumference?

A: If the arc length is $\frac{1}{8}$ of the circumference, then the central angle is $\frac{1}{8}$ of the total angle of the circle, which is 2π radians. Therefore, the radian measure of the central angle is $\frac{1}{4}$π radians.

Q: Can you give an example of how to use the formula θ = L/r to find the central angle if the arc length is $\frac{1}{8}$ of the circumference?

A: Yes, let's say we have a circle with a radius of 5 units, and an arc length of 2.5 units, which is $\frac{1}{8}$ of the circumference. We can use the formula to find the central angle:

θ = L/r θ = 2.5/5 θ = 0.5 radians

Q: What is the radian measure of the central angle if the arc length is $\frac{1}{16}$ of the circumference?

A: If the arc length is $\frac{1}{16}$ of the circumference, then the central angle is $\frac{1}{16}$ of the total angle of the circle, which is 2π radians. Therefore, the radian measure of the central angle is $\frac{1}{8}$π radians.

Q: Can you give an example of how to use the formula θ = L/r to find the central angle if the arc length is $\frac{1}{16}$ of the circumference?

A: Yes, let's say we have a circle with a radius of 5 units, and an arc length of 1.25 units, which is $\frac{1}{16}$ of the circumference. We can use the formula to find the central angle:

θ = L/r θ = 1.25/5 θ = 0.25 radians

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the radian measure of the central angle if the arc length is $\frac{1}{4}$ of the circumference. We hope that this article has been helpful in clarifying the relationship between arc length and central angles in circles.