Definition of Arc Length
Arc length is the distance along a curved line or a segment of the circumference of a circle. It measures the path traced along the curve, differentiating it from the straight-line distance between the arc's two endpoints.
Calculating Arc Length for a Circle
For a segment of a circle, the arc length (s) is typically calculated using the formula: s = rθ, where 'r' is the radius of the circle and 'θ' (theta) is the central angle subtended by the arc, measured in radians. If the angle is given in degrees, it must first be converted to radians before applying the formula.
Practical Example of Arc Length
Consider a swing moving in an arc. The distance a child travels from one end of the swing's path to the other, along the curve, represents the arc length. Another example is measuring the length of a piece of track for a toy train on a circular railway segment.
Importance and Advanced Applications
Arc length is a foundational concept in geometry, essential for understanding parts of circles and other curves. In more advanced mathematics, such as calculus, techniques involving integration are used to determine the arc length of complex, non-circular curves, finding applications in engineering design, navigation systems, and computer graphics.