An Arc On A Circle Measures 125 ∘ 125^{\circ} 12 5 ∘ . The Measure Of The Central Angle, In Radians, Is Within Which Range?A. 0 0 0 To Π 2 \frac{\pi}{2} 2 Π ​ Radians B. Π 2 \frac{\pi}{2} 2 Π ​ To Π \pi Π Radians C. Π \pi Π To

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Introduction

When dealing with circles and angles, it's essential to understand the relationship between degrees and radians. In this article, we will explore how to convert a central angle from degrees to radians and determine the range in which the measure of the central angle falls.

Understanding Degrees and Radians

Degrees and radians are two units used to measure angles. Degrees are commonly used in everyday life, while radians are used in mathematics and physics. To convert degrees to radians, we use the following formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Converting $125^{\circ}$ to Radians

Using the formula above, we can convert $125^{\circ}$ to radians:

$$\text{radians} = \frac{\pi}{180} \times 125$$

$$\text{radians} = \frac{125\pi}{180}$$

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5:

$$\text{radians} = \frac{25\pi}{36}$$

Understanding the Range of Radians

Now that we have converted $125^{\circ}$ to radians, we need to determine the range in which the measure of the central angle falls. To do this, we need to understand the relationship between the central angle and the arc length.

The Relationship Between Central Angle and Arc Length

The central angle is the angle formed by two radii that intersect at the center of the circle. The arc length is the length of the arc subtended by the central angle. The relationship between the central angle and the arc length is given by:

$$\text{arc length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the central angle in degrees, $r$ is the radius of the circle, and $2\pi r$ is the circumference of the circle.

Determining the Range of Radians

Since the arc length is directly proportional to the central angle, we can use the relationship above to determine the range of radians. We know that the arc length is $125^{\circ}$, which is equivalent to $\frac{125\pi}{180}$ radians. To determine the range of radians, we need to find the minimum and maximum values of the central angle.

Minimum and Maximum Values of Central Angle

The minimum value of the central angle is $0^{\circ}$, which is equivalent to $0$ radians. The maximum value of the central angle is $360^{\circ}$, which is equivalent to $2\pi$ radians.

Determining the Range of Radians

Using the relationship above, we can determine the range of radians:

$$\frac{0}{360} \times 2\pi r \leq \frac{125\pi}{180} \times 2\pi r \leq \frac{360}{360} \times 2\pi r$$

Simplifying the above inequality, we get:

$$0 \leq \frac{125\pi}{180} \times 2\pi r \leq 2\pi r$$

Conclusion

In conclusion, the measure of the central angle, in radians, is within the range of $\frac{25\pi}{36}$ to $\frac{2\pi}{3}$.

Final Answer

The final answer is $\boxed{\frac{25\pi}{36} \text{ to } \frac{2\pi}{3}}$.

Discussion

The discussion category for this article is mathematics. The article explores the relationship between degrees and radians and determines the range in which the measure of the central angle falls.

Related Articles

• Converting Degrees to Radians
• Understanding the Relationship Between Central Angle and Arc Length
• Determining the Range of Radians

References

• [1] "Degrees and Radians." Math Open Reference, mathopenref.com/anglesdegrees.html.
• [2] "Central Angle and Arc Length." Math Is Fun, mathisfun.com/geometry/central-angle-arc-length.html.
  

Introduction

In our previous article, we explored the relationship between degrees and radians and determined the range in which the measure of the central angle falls. In this article, we will answer some frequently asked questions about degrees and radians.

Q1: What is the difference between degrees and radians?

A1: Degrees and radians are two units used to measure angles. Degrees are commonly used in everyday life, while radians are used in mathematics and physics. The main difference between degrees and radians is the way they are defined. Degrees are defined as 1/360 of a circle, while radians are defined as the ratio of the arc length to the radius of a circle.

Q2: How do I convert degrees to radians?

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

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Q3: How do I convert radians to degrees?

A3: To convert radians to degrees, you can use the following formula:

$$\text{degrees} = \frac{180}{\pi} \times \text{radians}$$

Q4: What is the relationship between the central angle and the arc length?

A4: The central angle is the angle formed by two radii that intersect at the center of the circle. The arc length is the length of the arc subtended by the central angle. The relationship between the central angle and the arc length is given by:

$$\text{arc length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the central angle in degrees, $r$ is the radius of the circle, and $2\pi r$ is the circumference of the circle.

Q5: How do I determine the range of radians?

A5: To determine the range of radians, you need to find the minimum and maximum values of the central angle. The minimum value of the central angle is $0^{\circ}$, which is equivalent to $0$ radians. The maximum value of the central angle is $360^{\circ}$, which is equivalent to $2\pi$ radians.

Q6: What is the range of radians for a central angle of $125^{\circ}$?

A6: Using the relationship above, we can determine the range of radians for a central angle of $125^{\circ}$:

$$\frac{0}{360} \times 2\pi r \leq \frac{125\pi}{180} \times 2\pi r \leq \frac{360}{360} \times 2\pi r$$

Simplifying the above inequality, we get:

$$0 \leq \frac{125\pi}{180} \times 2\pi r \leq 2\pi r$$

Q7: What is the final answer for the range of radians?

A7: The final answer for the range of radians is $\boxed{\frac{25\pi}{36} \text{ to } \frac{2\pi}{3}}$.

Q8: What is the relationship between degrees and radians in terms of the unit circle?

A8: The unit circle is a circle with a radius of 1. The relationship between degrees and radians in terms of the unit circle is that 1 degree is equal to $\frac{\pi}{180}$ radians, and 1 radian is equal to $\frac{180}{\pi}$ degrees.

Q9: How do I use the unit circle to convert degrees to radians?

A9: To use the unit circle to convert degrees to radians, you can use the following formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Q10: How do I use the unit circle to convert radians to degrees?

A10: To use the unit circle to convert radians to degrees, you can use the following formula:

$$\text{degrees} = \frac{180}{\pi} \times \text{radians}$$

Conclusion

In conclusion, we have answered some frequently asked questions about degrees and radians. We have explored the relationship between degrees and radians and determined the range in which the measure of the central angle falls. We have also used the unit circle to convert degrees to radians and radians to degrees.

Final Answer

The final answer is $\boxed{\frac{25\pi}{36} \text{ to } \frac{2\pi}{3}}$.

Discussion

The discussion category for this article is mathematics. The article explores the relationship between degrees and radians and answers some frequently asked questions.

Related Articles

• Understanding the Relationship Between Degrees and Radians
• Converting Degrees to Radians
• Converting Radians to Degrees

References

• [1] "Degrees and Radians." Math Open Reference, mathopenref.com/anglesdegrees.html.
• [2] "Central Angle and Arc Length." Math Is Fun, mathisfun.com/geometry/central-angle-arc-length.html.