Arc Length & Sector
Arc length, sector area and chord from a radius and an angle, in degrees or radians.
θ = 1.047198 rad · the sector is 16.667% of the full circle
An arc is the fraction of a circle’s circumference that the angle cuts out, and a sector is the matching fraction of its area. In radians both are one line: arc = rθ and sector area = ½r²θ.
Why radians make the formula disappear
A radian is defined as the angle that cuts an arc exactly as long as the radius. That definition is the whole formula: an angle of θ radians cuts θ radii of arc, so the arc is rθ with nothing else attached. The degree version needs a conversion factor — θ ÷ 360 × 2πr — for no reason other than that 360 is an arbitrary number inherited from Babylonian astronomy.
The sector area follows the same logic. A sector is the fraction θ ÷ 2π of the full circle, so its area is that fraction of πr², which simplifies to ½r²θ. Both results are proportions, which is why the answer scales linearly with the angle and quadratically with the radius.
The chord is a different object
The chord is the straight line between the arc’s endpoints, not a piece of the circle, so it needs trigonometry rather than a proportion: 2r·sin(θ ÷ 2). It is always shorter than the arc, and the gap widens as the angle grows — at 60° the chord equals the radius exactly, which is why an inscribed hexagon has sides equal to its circumradius.
θ in radians; in degrees multiply by π ÷ 180 first
- 1 Put the angle in radians. 60° × π ÷ 180 = π/3 ≈ 1.047198 radians.
- 2 Multiply by the radius for the arc. With r = 5: 5 × 1.047198 = 5.235988, which is 5π/3.
- 3 Use ½r²θ for the sector area. ½ × 25 × 1.047198 = 13.089969, which is 25π/6.
- 4 Use 2r·sin(θ ÷ 2) for the chord. 2 × 5 × sin(30°) = 10 × 0.5 = 5 exactly — the chord equals the radius at 60°.
- 5 Sanity-check against the fraction. 60° is one sixth of the circle, so the arc should be one sixth of 2π × 5 = 31.415927, and 31.415927 ÷ 6 = 5.235988. It matches.
Common angles at radius 1
Multiply the arc and chord by r, and the sector area by r², for any other radius.
| Angle | Radians | Arc | Sector area | Chord |
|---|---|---|---|---|
| 30° | π/6 | 0.523599 | 0.261799 | 0.517638 |
| 45° | π/4 | 0.785398 | 0.392699 | 0.765367 |
| 60° | π/3 | 1.047198 | 0.523599 | 1 |
| 90° | π/2 | 1.570796 | 0.785398 | 1.414214 |
| 120° | 2π/3 | 2.094395 | 1.047198 | 1.732051 |
| 180° | π | 3.141593 | 1.570796 | 2 |
| 360° | 2π | 6.283185 | 3.141593 | 0 |
Minor and major, and where this gets used
Any two points on a circle cut it into two arcs. The smaller is the minor arc and the larger the major arc, and they add to the full circumference — so the major arc for a 60° angle is the 300° one. If a problem gives you an arc and asks for the angle, check which of the two you have been handed before dividing by the radius.
Beyond geometry homework, the same arithmetic is what converts a wheel’s rotation into distance travelled, sizes a curved road or a pipe bend, and turns an angular measurement in astronomy into a physical size. In each case the radius carries the units and the angle in radians is a pure number, which is why arc length comes out in whatever units the radius was in.
One boundary worth naming: this is the arc of a circle. The arc length of a general curve needs the integral ∫√(1 + (dy/dx)²) dx, which is a calculus problem rather than a proportion, and not what this page does.