Arc Length Calculator

For a circle of radius r and central angle theta measured in radians, the arc length s equals r times theta. If your angle is in degrees, convert first by multiplying by pi / 180. For example, a 90 degree arc on a circle of radius 5 has length 5 times pi / 2, which is about 7.854 units.

The arc length follows the curve along the edge of the circle; the chord is the straight line between the two endpoints of the arc. The chord is always shorter than the arc. The chord formula is c = 2 r sin(theta / 2), so for r = 5 and theta = 90 degrees the chord is about 7.071, while the arc is about 7.854.

The sagitta h is the perpendicular distance from the midpoint of the chord up to the midpoint of the arc. It equals r times (1 minus cos(theta / 2)). You can also recover the radius from a chord c and sagitta h using r = (c squared / 8h) plus (h / 2), which is handy in carpentry, masonry, and bridge layout.

The sector area is the pie slice bounded by two radii and the arc: (1 / 2) r squared theta. The segment area is the smaller region between the chord and the arc: (1 / 2) r squared (theta minus sin theta). For r = 5 and theta = pi / 2 the sector area is about 19.635 and the segment area is about 7.135.

Not directly. For a general curve y = f(x), the arc length from x = a to x = b is the integral of sqrt(1 plus (dy / dx) squared) dx. That requires calculus and usually a numerical method. This calculator covers the common circular case in closed form. For other shapes, use a symbolic or numerical integrator.