Circular sector calculator
Sector area, arc length and chord from the radius and the angle.
Fill in the fields and the result will appear here automatically.
A sector is specified by a positive radius and a central angle greater than 0° and at most 360°. Results include area, arc, chord, perimeter and circle share. An incomplete sector’s boundary has an arc and two radii; at 360° the shape is the whole disc and its perimeter is just the circumference. The full-circle chord is geometrically zero; small positive chords are not forced to zero.
How it works
Formula and logic
The angle becomes radians as θ = α·π/180. The sector area is S = ½r²θ, the arc length L = rθ and the chord c = 2r·sin(θ/2). For 0° < α < 360°, perimeter is L+2r; at α = 360° it is L = 2πr. The circle fraction is α/360. Formulas use radians internally, while input and angle labels stay in degrees.
Example
A sector of radius 5 cm with a 60° angle has an area of 13.09 cm², an arc of 5.236 cm and a chord of exactly 5 cm.
Fields and units
- Length unit — list option
- Radius — cm
- Central angle — °
How to use
- — Pick the length unit.
- — Enter the radius.
- — Give the central angle in degrees.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: circle area and circumference OpenStax: sine and tangent in a right triangle
- Limitation
- Area describes a sector, not the segment between arc and chord. At a full circle the perimeter is the outer boundary without a radial cut. Radius uses the selected unit; switching units does not convert its number.
FAQ
Why is the chord zero at 360 degrees?
Because the ends of the arc coincide: the segment joining them collapses to a point. Binary arithmetic leaves a tiny residue there, and it is deliberately snapped to exact zero.
How does a chord differ from the arc length?
The arc follows the circle, the chord runs straight between its ends. The chord is always shorter, and the gap widens with the angle.
Why convert degrees to radians?
Because S = ½r²θ and L = rθ only hold in radian measure. Substituting degrees would be out by a factor of about 57.
How do I get the area of a segment?
For the corresponding oriented arc, segment area is ½r²(θ−sinθ), with θ in radians. For θ ≤ π the central triangle area is subtracted; for θ > π sinθ is negative and the formula gives the major segment. This tool displays a sector, not a segment.