Arc Length Circular

Definition

Arc length ($s$) is the distance along the curved edge of a circle subtended by a central angle.

Why It Matters

It establishes the natural link between angular rotation and physical distance, which is fundamental to mechanical engineering and physics. This connection allows for the design of everything from gears to planetary orbits.

Core Concepts

• Formula using Radians: $s = r\theta$ where $r$ is the radius and $\theta$ is the central angle in radians.

  • How to read: “s equals r times theta.”
  • Meaning: Direct formula when angle is in radians—arc length scales linearly with both radius and angle.

• Formula using Degrees: $s = \frac{\theta^\circ}{360^\circ} \cdot 2\pi r$

  • How to read: “s equals theta in degrees divided by three hundred sixty degrees, times two pi r.”
  • Meaning: Fraction of a full circumference ($2\pi r$) corresponding to the angle’s share of $360^\circ$.

• Linearity: The arc length is directly proportional to both the radius and the angle of rotation.

Connected Concepts

• Angles and Radian Measure
• Sector Area Circular
• Linear Speed
• Angular Speed