Math
Arc Length Calculator
Calculate the arc length, chord, sagitta, sector area, and segment area of a circular arc from radius and central angle, arc, or chord.
Circular arc inputs
Arc length
7.854
- Central angle (rad)
- 1.570796
- Central angle (deg)
- 90
- Chord length (c)
- 7.0711
- Sagitta (h)
- 1.4645
- Sector area
- 19.635
- Segment area
- 7.135
- Radius (r)
- 5
General arc length (calculus form)
For an arbitrary curve y = f(x) between x = a and x = b, the arc length is
L = integral from a to b of sqrt(1 + (dy/dx)^2) dx
This calculator does not evaluate that integral. Use the circular arc inputs above for closed-form results.
Frequently Asked Questions about the Arc Length Calculator
How do you calculate the arc length of a circle?
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.
What is the difference between arc length and chord length?
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.
How is the sagitta (arc height) related to the radius and chord?
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.
What is the sector area versus the segment area?
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.
Can this calculator handle non-circular curves?
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.