Finding The Radius Of A Circle With A Given Central Angle And Arc Length

In the realm of geometry, circles hold a fundamental place, and understanding their properties is crucial for various mathematical and real-world applications. One such property is the relationship between the central angle, the arc length it intercepts, and the circle's radius. In this article, we will delve into the process of determining the radius of a circle given the central angle and the intercepted arc length. Specifically, we will tackle the problem of finding the radius of a circle where a central angle measuring $\frac{5 \pi}{6}$ radians intercepts an arc on the circle with a length of $35 \pi$ centimeters. This exploration will not only provide a step-by-step solution to this particular problem but also enhance our understanding of the underlying principles governing circles and their properties.

Understanding the Relationship Between Central Angle, Arc Length, and Radius

The relationship between the central angle, arc length, and radius of a circle is a cornerstone of circular geometry. It provides a direct link between the angle formed at the center of the circle, the portion of the circle's circumference that the angle encompasses (the arc length), and the distance from the center to any point on the circle (the radius). This relationship is mathematically expressed by the formula:

$\qquad s = r \theta$

where:

• s represents the arc length, which is the distance along the curved edge of the circle intercepted by the central angle.
• r denotes the radius of the circle, the distance from the center to any point on the circumference.
• $\theta$ signifies the central angle measured in radians, which is the angle formed at the center of the circle by the two radii that connect the endpoints of the arc.

This formula elegantly captures the proportionality between these three quantities. It states that the arc length is equal to the product of the radius and the central angle in radians. This means that if you know any two of these quantities, you can readily calculate the third. For instance, if you have the radius and the central angle, you can determine the arc length. Conversely, if you have the arc length and the radius, you can find the central angle, and as we will see in this article, if you have the arc length and the central angle, you can calculate the radius. Understanding this fundamental relationship is key to solving various problems related to circles, including the one we will address in this article.

Problem Statement: Finding the Radius

Our mission is to find the radius of a circle based on the following information:

• Central Angle: The central angle, denoted by $\theta$, measures $\frac{5 \pi}{6}$ radians. This angle originates from the circle's center and spans a portion of the circle's circumference.
• Arc Length: The arc length, represented by s, is $35 \pi$ centimeters. This is the distance along the circle's curved edge that the central angle intercepts.

To solve this problem, we will employ the fundamental relationship between the central angle, arc length, and radius, which, as we discussed earlier, is expressed by the formula:

$\qquad s = r \theta$

Our goal is to determine the radius (r). To achieve this, we will rearrange the formula to isolate r on one side of the equation. This algebraic manipulation will allow us to directly calculate the radius using the given values of the arc length and central angle. The rearranged formula will serve as our primary tool in solving this problem and uncovering the circle's radius.

Solution: Applying the Formula and Solving for the Radius

To find the radius (r) of the circle, we will utilize the formula that connects arc length (s), radius (r), and central angle ($\theta$):

$\qquad s = r \theta$

We are given the arc length s as $35 \pi$ centimeters and the central angle $\theta$ as $\frac{5 \pi}{6}$ radians. Our objective is to isolate r in the equation. To do this, we will divide both sides of the equation by $\theta$:

$\qquad r = \frac{s}{\theta}$

Now, we can substitute the given values into the rearranged formula:

$\qquad r = \frac{35 \pi}{\frac{5 \pi}{6}}$

To simplify this expression, we need to divide by the fraction $\frac{5 \pi}{6}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{5 \pi}{6}$ is $\frac{6}{5 \pi}$. So, we have:

$\qquad r = 35 \pi \cdot \frac{6}{5 \pi}$

Now, we can simplify by canceling out the common factors. The $\pi$ in the numerator and denominator cancel out, and 35 and 5 have a common factor of 5. Dividing 35 by 5 gives 7:

$\qquad r = 7 \cdot 6$

Finally, we multiply 7 by 6 to obtain the radius:

$\qquad r = 42 \text{ cm}$

Therefore, the radius of the circle is 42 centimeters.

Answer and Conclusion

Based on our calculations, the radius of the circle is 42 cm. This corresponds to option B in the given choices. This problem highlights the practical application of the relationship between central angles, arc lengths, and radii in circles. By understanding and applying the formula $s = r \theta$, we can solve various geometric problems involving circles.

In conclusion, we successfully determined the radius of the circle by utilizing the fundamental relationship between the arc length, central angle, and radius. This problem serves as a valuable exercise in applying geometric principles and algebraic manipulation to solve real-world problems. The key takeaway is the importance of understanding and utilizing the formula $s = r \theta$ when dealing with circles and their properties. This knowledge is crucial not only for academic pursuits but also for practical applications in fields such as engineering, architecture, and design, where circular geometry plays a significant role.

Practice Problems

To further solidify your understanding of the concepts discussed, here are a few practice problems:

1. A central angle of $\frac{2\pi}{3}$ radians intercepts an arc of length $16\pi$ cm. Find the radius of the circle.
2. A circle has a radius of 10 cm. What is the arc length intercepted by a central angle of $\frac{3\pi}{4}$ radians?
3. An arc length of 25 cm is intercepted by a central angle of $\frac{5\pi}{6}$ radians. What is the radius of the circle?

Attempting these problems will reinforce your understanding of the relationship between central angles, arc lengths, and radii, and further enhance your problem-solving skills in circular geometry. Remember to utilize the formula $s = r\theta$ and apply algebraic manipulation techniques to solve for the unknown variable. These practice problems will not only prepare you for similar questions but also provide a deeper appreciation for the elegance and practicality of circular geometry principles.

By working through these problems, you'll gain confidence in your ability to tackle various geometric challenges and develop a more intuitive understanding of circles and their properties. Don't hesitate to review the solution steps outlined in this article if you encounter any difficulties, and remember that consistent practice is key to mastering any mathematical concept.