ConverterHub

Circular Sector Calculator

Find the arc length and area of a sector.

Result

15.71 units
78.54 units²

How it is calculated

A sector is a slice of a circle between two radii. The angle is entered in degrees and converted to radians with θ = degrees × π / 180. Then arc length = r × θ and area = ½ × r² × θ. These are the same as the fraction-of-circle shortcuts 2πr × (angle / 360) and πr² × (angle / 360). The radius must be above zero and the angle above 0 and at most 360. Results have two decimals, in the length unit of your radius (arc) and its square (area). The tool does not include the two straight edges; add 2r for the full perimeter of the slice.

Worked example

A sector with radius 10 and a 90° angle. Convert the angle: 90 × π / 180 = 1.5708 rad. Arc length = 10 × 1.5708 = 15.71 units. Area = ½ × 10² × 1.5708 = 78.54 units². This is exactly a quarter of the full circle, whose circumference is 62.83 and area is 314.16. For the perimeter of the slice, add the two radii: 15.71 + 20 = 35.71.

Try this example →

Common mistakes

  • Entering the angle in radians; the field is in degrees, and 1.57 would be treated as under 2 degrees.
  • Using the diameter in place of the radius, which makes the area four times too large.
  • Taking arc length as the sector's perimeter; the two straight edges add 2r.
  • Typing an angle over 360 for a spiral or multiple turns; values above 360 are rejected.
  • Mixing units, such as radius in cm and expecting the area in m².

About this calculator

A circular sector is the pie-slice-shaped region bounded by two radii and the arc between them. This calculator finds the arc length, s = r·θ, and the sector area, A = ½·r²·θ, where θ is the central angle measured in radians (the calculator handles degree input by converting internally).

These formulas come directly from the definition of a radian — the angle that produces an arc length equal to the radius — and scale down proportionally from the full circle's circumference (2πr) and area (πr²) by the fraction of the full angle (θ/2π) the sector represents.

Engineers and machinists calculating material for curved parts like gears, pipe fittings, or sheet-metal cutouts, architects designing curved features like fan windows or arches, and geometry students working through circle problems all use this calculator by entering a radius and central angle.

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