Arc Length

Algebra-2

1. Fundamental Concepts

Definition: The arc length of a circle is the distance along the curved line making up the arc within the circle.
Formula: The formula for the arc length \(s\) of a circle with radius \(r\) and central angle \(\theta\) (in radians) is given by \(s = r\theta\).
Conversion: To convert degrees to radians, use the formula \(\theta_{\text{radians}} = \frac{\pi}{180} \cdot \theta_{\text{degrees}}\).

2. Key Concepts

Basic Rule: $$s = r\theta$$

Degree to Radian Conversion: $$\theta_{\text{radians}} = \frac{\pi}{180} \cdot \theta_{\text{degrees}}$$

Application: Used in various fields including physics, engineering, and geometry to calculate distances along circular paths.

3. Examples

Example 1 (Basic)

Problem: Find the arc length of a circle with a radius of 5 units and a central angle of 60 degrees.

Step-by-Step Solution:

Convert the angle from degrees to radians: \(\theta_{\text{radians}} = \frac{\pi}{180} \cdot 60 = \frac{\pi}{3}\).
Calculate the arc length using the formula \(s = r\theta\): \(s = 5 \cdot \frac{\pi}{3} = \frac{5\pi}{3}\).

Validation: Substitute values into the formula → Original: \(s = 5 \cdot \frac{\pi}{3}\); Simplified: \(\frac{5\pi}{3}\) ✓

Example 2 (Intermediate)

Problem: A circle has an arc length of 10 units and a central angle of \(\frac{\pi}{4}\) radians. Find the radius of the circle.

Step-by-Step Solution:

Use the arc length formula \(s = r\theta\) and solve for \(r\): \(10 = r \cdot \frac{\pi}{4}\).
Rearrange to find \(r\): \(r = \frac{10}{\frac{\pi}{4}} = \frac{10 \cdot 4}{\pi} = \frac{40}{\pi}\).

Validation: Substitute values into the formula → Original: \(10 = r \cdot \frac{\pi}{4}\); Simplified: \(r = \frac{40}{\pi}\) ✓

4. Problem-Solving Techniques

Visual Strategy: Draw the circle and label the radius, central angle, and arc length to visualize the problem.
Error-Proofing: Always check units and ensure that angles are in the correct form (radians or degrees as required).
Concept Reinforcement: Practice converting between degrees and radians to ensure accuracy in calculations.