5 cm radius, 60° angle
Arc = 5π/3 cm
Arc length is approximately 5.235988 cm. Area is 25π/6, or approximately 13.089969 cm². The chord is exactly 5 cm; adding both radii to the arc gives perimeter ≈ 15.235988 cm.
Geometry and measurement
Circle Sector Calculator
Enter a radius and central angle to measure a circular sector, from a narrow wedge to a full circle.
Calculate circle sector measurements
Arc length
Enter a radius and central angle, then select Calculate.
- Sector area
- Chord length
- Sector perimeter
Calculations run in your browser. Press Enter to calculate. Successful results appear in this tab's Recent calculations. Measurement uncertainty is not removed by displaying more decimals.
Worked examples
4 m radius, 1.5 rad angle
Arc = 4 × 1.5 = 6 m
Sector area is ½ × 4² × 1.5 = 12 m². Perimeter is 6 + 8 = 14 m. The chord is 8 × sin(0.75), approximately 5.45311 m.
3 in radius, 360° angle
Arc = perimeter = 6π in
Both equal approximately 18.849556 in. Area is 9π, approximately 28.274334 in². The endpoints coincide, so the chord is 0 in. The full circle's boundary has no extra radial edges.
Formulas and precision
For radius r and central angle θ in radians, arc length is rθ, sector area is r²θ/2, and chord length is 2r sin(θ/2). Convert degrees into radians by multiplying by π/180. Area uses square units; the other measurements use the radius unit.
For 0 < θ < 2π, the sector perimeter is rθ + 2r. This tool treats exactly 360° or 2π as a whole disk, whose perimeter is 2πr. A nearly complete sector still has two radial edges; a complete disk does not. This boundary convention is deliberate.
Use pi or π for exactly half a turn in radians, and 2pi or 2π for exactly one turn. Other expressions, such as pi/2 or 2*pi, are not accepted. Use 90 degrees for an exact quarter turn. Numeric radians are used as typed: 6.2831853071 is just below 2π, while 6.2831853072 is above it and is rejected. No nearby decimal is silently changed to 2π.
Calculations use a separate 120-significant-digit decimal context. π and sine are numerical approximations; exact whole-circle and selected chord identities are handled directly. Only the final displayed values are rounded, to at most 0, 2, 4, 6, or 10 decimal places. Exact halfway values round up, and trailing zeros are removed. The ≈ symbol marks numerical approximation or display rounding. A positive value that displays zero receives an explanatory notice.
References: OpenStax: Angles for radians, arc length, and sector area; OpenStax: Sine and Cosine Functions for trigonometric definitions. The chord relationship follows by splitting the isosceles triangle into two right triangles, using the smaller angle between the radii. See our calculation policy for shared input and rounding rules.
Common questions
Is a sector the same as a circular segment?
No. A sector is bounded by two radii and an arc. A circular segment is bounded by a chord and an arc. This calculator reports sector area, not the area between the chord and the curve.
Can I calculate an angle larger than 180 degrees?
Yes, up to 360 degrees or 2π radians. For example, 300 degrees describes a major sector. Its chord equals that of a 60-degree sector with the same radius, but its arc and area are five times as large.
Why does the perimeter change at exactly one full turn?
A sector just below 360 degrees has a narrow missing wedge. Its boundary includes two radial edges. At exactly 360 degrees, the wedge closes and this tool measures the boundary of the complete disk, which is only its circumference. It does not count an imaginary cut through the disk.
Can I enter a zero or negative angle?
No. This tool measures a nonzero sector from a positive sweep of at most one revolution. Negative angles and multiple turns can describe rotations, but they do not identify one sector under this tool's rules.
Why is my chord nearly zero but not exactly zero?
A numeric radian value close to 2π still represents a sector with distinct endpoints. Its chord is small and positive. Enter 2pi in radians or 360 in degrees for an exact full circle. Increasing display precision can reveal a small value currently rounded to zero.