Calculating Circle Radius Given Arc Length And Central Angle

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In the realm of geometry, circles hold a special place, and understanding their properties is crucial for various mathematical applications. One fundamental aspect of a circle is the relationship between its radius, central angle, and arc length. This article delves into a problem that involves calculating the radius of a circle given the measure of a central angle and the length of the corresponding minor arc. Let's explore the concepts and steps involved in solving this problem.

Problem Statement

Consider a circle denoted as $C$. Within this circle, we have an angle $\angle ACB$ that measures $\frac{3 \pi}{4}$ radians. This angle is subtended by a minor arc $AB$, which has a length of $9 \pi$ inches. Our objective is to determine the length of the radius of circle $C$. If necessary, we will round our answer to the nearest tenth.

Understanding the Concepts

Before we embark on the solution, it's essential to grasp the underlying concepts that govern the relationship between angles, arcs, and radii in a circle.

Central Angle

A central angle is an angle whose vertex lies at the center of the circle, and its sides intersect the circle at two distinct points. The measure of a central angle is directly related to the length of the arc it subtends.

Arc Length

An arc is a portion of the circumference of a circle. The length of an arc is the distance along the curved path of the arc. The length of an arc is proportional to the central angle that subtends it.

Relationship between Central Angle, Arc Length, and Radius

The fundamental relationship that connects the central angle, arc length, and radius of a circle is given by the formula:

s=rthetas = r \\theta

where:

  • s$ is the arc length

  • r$ is the radius of the circle

  • \\theta$ is the central angle in radians

This formula states that the arc length is equal to the product of the radius and the central angle (in radians).

Solving the Problem

Now that we have a clear understanding of the concepts, let's apply them to solve the problem at hand.

  1. Identify the given information:

    • Central angle, $\theta = \frac{3 \pi}{4}$ radians
    • Arc length, $s = 9 \pi$ inches

  2. Apply the formula:

    We know that $s = r \theta$. We need to find the radius $r$, so we can rearrange the formula as follows:

    r=fracsthetar = \\frac{s}{\\theta}

  3. Substitute the values:

    Substitute the given values of $s$ and $\theta$ into the formula:

    r=frac9pifrac3pi4r = \\frac{9 \\pi}{\\frac{3 \\pi}{4}}

  4. Simplify the expression:

    To divide by a fraction, we multiply by its reciprocal:

    r=9picdotfrac43pir = 9 \\pi \\cdot \\frac{4}{3 \\pi}

    The $\pi$ terms cancel out:

    r=9cdotfrac43r = 9 \\cdot \\frac{4}{3}

    Simplify further:

    r=frac363r = \\frac{36}{3}

    r=12r = 12

  5. State the answer:

    Therefore, the length of the radius of circle $C$ is 12 inches.

Alternative Approach: Proportions

Another way to think about this problem is to use proportions. The ratio of the arc length to the circumference of the circle is equal to the ratio of the central angle to the total angle in a circle (which is $2\pi$ radians).

Circumference of a Circle

Recall that the circumference of a circle is given by the formula:

C=2pirC = 2 \\pi r

where $r$ is the radius of the circle.

Setting up the Proportion

We can set up the following proportion:

fractextArcLengthtextCircumference=fractextCentralAngletextTotalAngle\\frac{\\text{Arc Length}}{\\text{Circumference}} = \\frac{\\text{Central Angle}}{\\text{Total Angle}}

Substituting the given values and the formula for the circumference, we get:

frac9pi2pir=fracfrac3pi42pi\\frac{9 \\pi}{2 \\pi r} = \\frac{\\frac{3 \\pi}{4}}{2 \\pi}

Solving the Proportion

To solve for $r$, we can cross-multiply:

9picdot2pi=frac3pi4cdot2pir9 \\pi \\cdot 2 \\pi = \\frac{3 \\pi}{4} \\cdot 2 \\pi r

Simplify:

18pi2=frac3pi22r18 \\pi^2 = \\frac{3 \\pi^2}{2} r

Multiply both sides by $\frac{2}{3 \pi^2}$:

r=18pi2cdotfrac23pi2r = 18 \\pi^2 \\cdot \\frac{2}{3 \\pi^2}

The $\pi^2$ terms cancel out:

r=18cdotfrac23r = 18 \\cdot \\frac{2}{3}

r=frac363r = \\frac{36}{3}

r=12r = 12

This approach also yields the same answer: the radius of circle $C$ is 12 inches.

Key Takeaways

This problem illustrates the fundamental relationship between the central angle, arc length, and radius of a circle. By understanding and applying the formula $s = r \theta$, we can solve a variety of problems involving circles. Additionally, the use of proportions provides an alternative approach to solving such problems.

In summary, to determine the radius of a circle when given the arc length and central angle:

  • First, it's essential to understand the formula that connects these elements, which is $s = r \theta$, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians.
  • Second, rearrange the formula to solve for the radius: $r = s / \theta$.
  • Third, substitute the given values for the arc length and central angle.
  • Finally, simplify the expression to find the radius. Remember, practice and a clear understanding of the underlying principles are key to mastering these concepts.

Conclusion

In conclusion, we have successfully determined the radius of circle $C$ to be 12 inches by utilizing the relationship between the central angle, arc length, and radius. This problem serves as a valuable exercise in applying geometric principles and reinforces the importance of understanding the connections between different elements of a circle. Whether you use the direct formula or proportions, the key is to apply the concepts correctly and with precision. Keep practicing, and you'll find these geometric calculations becoming second nature.