About Arc Length
What Is Arc Length?
Arc length is the distance along a portion of the circumference of a circle. It is determined by the radius of the circle and the central angle that subtends the arc. Arc length is fundamental in geometry, engineering, and physics.
The Formula
The arc length formula is L = r x theta, where r is the radius and theta is the central angle in radians. For degrees, the formula becomes L = r x theta x (pi/180). The sector area is A = (1/2)r² x theta (radians) or A = (1/2)r² x theta x (pi/180) (degrees).
Related Measurements
From the radius and angle, you can also calculate the chord length (straight line between arc endpoints) as c = 2r x sin(theta/2), and the segment height (distance from chord to arc midpoint) as h = r(1 - cos(theta/2)). These are all interconnected.
Applications
Arc length calculations are essential in road and railway design (curves), architecture (arches), manufacturing (bent materials), robotics (joint angles to distances), and navigation. Understanding arc properties allows precise design of curved structures.