Circular Arc Properties and Relationships

A circular arc is a portion of the circumference of a circle. Understanding its properties and relationships to other elements of the circle, such as the central angle and the circle's perimeter, is crucial in geometry, engineering, and other fields.

Arc Length and Central Angle

The length of an arc is directly proportional to the measure of its central angle. The central angle is the angle formed at the circle's center by the two radii that connect the endpoints of the arc to the center. A full circle's central angle measures 360 degrees (or 2π radians).

Arc Length Formula

The fundamental relationship between arc length (s), central angle (θ in radians), and the circle's radius (r) is expressed as:

s = rθ

If the central angle is given in degrees (α), it must be converted to radians before using the formula. Conversion is achieved using the following relationship:

θ = (α π) / 180

Relating Arc Length to Circumference

The arc length is a fraction of the circle's total circumference. The circumference (C) of a circle is given by:

C = 2πr

The ratio of the arc length (s) to the circumference (C) equals the ratio of the central angle (θ in radians) to 2π (full circle in radians), or the ratio of the central angle (α in degrees) to 360° (full circle in degrees):

s / C = θ / 2π or s / C = α / 360°

Applications

• Engineering: Calculating the length of curved structural elements.
• Navigation: Determining distances along curved paths.
• Computer Graphics: Rendering and manipulating curved shapes.
• Manufacturing: Precision cutting and shaping of curved components.