Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 27 meters Central Angle \(\theta\) \(120^{\circ}\)

Step-by-Step Solution

Answer

The length of the arc intercepted by a central angle of \(120^{\circ}\) on a circle of radius 27 meters is \(L = 27 * (120 * \frac{\pi}{180})\) meters. This value can be further simplified by calculation.

Step 1: Convert the central angle from degrees to radians

To convert from degrees to radians, the formula is \(\theta_{radians} = \theta_{degrees} * \frac{\pi}{180}\). In this exercise the given central angle \(\theta\) is \(120^{\circ}\). So the conversion will be \(\theta_{radians} = 120 * \frac{\pi}{180}\)

Step 2: Substitute values into the formula

Now, we use the general formula for obtaining the arc length that is \(L = r * \theta\). Substituting the given radius and the converted angle values into the formula, we get \(L = 27 * (120 * \frac{\pi}{180})\)

Step 3: Solve the equation

Perform the multiplication operation to determine the length of the arc.

Key Concepts

Central Angle

A central angle in circle geometry refers to an angle whose vertex is located at the center of the circle. This angle subtends an arc on the circle, which simply means it "opens up" to span a portion of the circle's circumference. The length of this arc is directly related to the size of the central angle and the radius of the circle. In this context, if the angle is larger, the arc tends to be longer as well. In exercises involving central angles, students are often required to calculate the arc length or the angle itself when given other conditions or measurements. It is important to remember that the units of measurement for angles can affect how the central angle interacts with formulas, such as needing to convert degrees to radians.

Conversion Between Degrees and Radians

Since mathematical formulas involving circles often use radians, knowing how to convert between degrees and radians is crucial. Radians and degrees are two units for measuring angles, where calculators typically operate with radian measures. To convert an angle from degrees to radians, use the formula:

• \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\)

This conversion is necessary in many geometric calculations because radians provide a more natural measure when assessing angular distances, particularly in conjunction with circle properties. For instance, converting \(120^{\circ}\) into radians is simply multiplying by \(\frac{\pi}{180}\), resulting in \(\theta_{radians} = 2\pi/3\). Always ensure that you convert angles before substituting them in formulas involving radius and arc length calculations.

Circle Geometry

Circle geometry involves the study of properties and measurements related to circles. Understanding the interaction between these properties is essential to solving problems related to circles, such as calculating arc length or sector area. Key components include:

• Radius: the distance from the center to any point on the circumference. It's a fundamental measure in calculating other properties, such as diameter or circumference.
• Arc Length: the portion of the circumference that is "under" a given angle (central angle). Calculating arc length involves using the formula: \[L = r \times \theta_{radians}\]where \(L\) is the arc length, \(r\) is the radius, and \(\theta_{radians}\) is the central angle in radians.

Circle geometry provides the framework needed to tackle equations and validate measurements, making it pivotal in various practical and theoretical applications. For example, knowing the formula \(L = 27 \times (120 \times \frac{\pi}{180})\) allows us to find the arc length when \(r\) is 27 meters and the angle is \(120^{\circ}\). This approach seamlessly ties together radius measures, angle conversions, and the understanding of arc relationships within circle geometry.